Mesopotamian 6x9 by Raymond N. Shekoury

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MESOPOTAMIANS: PIONEERS Of MATHEMATICS By RAYMOND N. SHEKOURY Professor Emeritus of Mathematics University of Baghdad / Iraq

B L O O M I N G T O N

I L L I N

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O I S A P R I L

2005

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Table of Contents Preface ........................................................................................................ii .................................................................................................................... 2 01. Prehistoric People..................................................................................1 01.1. Introduction .................................................................................................... 1 01.2. The Early Stone-Age....................................................................................... 2 01.3. The Middle Stone-Age .................................................................................... 5 01.4. Emergence of Settlements .............................................................................. 7 01.5. Emergence of Administration ..................................................................... 10

0.2. Mesopotamia(ns) ................................................................................ 13 02.1. The Geography of Mesopotamia ................................................................ 13 02.2. The Cradle of Civilization ........................................................................... 17 02.3. A Very Brief History of the Mesopotamians ............................................ 19 02.4. Different Civilizations in Mesopotamia..................................................... 20 02.5. Grouping the Mesopotamian Civilizations Under a................................ 21 Single Name ............................................................................................................. 21 02.6. Cultural Sturdiness of Mesopotamian Civilizations................................ 23

03. Mesopotamian “CD’S”........................................................................ 25 03.1. Introduction ................................................................................................... 25 03.2. Social Need for Writings .............................................................................. 27

v 03.3. First Stage of Writing ................................................................................... 29 03.4. Media of Communication ............................................................................ 33 03.5. Second Stage of Writings ............................................................................. 35 03.6. Libraries ......................................................................................................... 36 03.7. Use of Cuneiform Writings in School-Education..................................... 37 03.8. Deciphering of the Cuneiform Writings.................................................... 39

04. Numbers, Numerals & Number-Systems............................................. 45 04.1. Introduction ................................................................................................... 45 04.2. Problems of the Emergence of the Notions of Numbers ......................... 49 04.3. Initiating the Modern Child in the Notion of NumberS ......................... 52 04.4. The gradual Discovery by the Ancients of the Notion of Numbers....... 58 04.5. The Place-Value System ............................................................................... 68 04.6. Truncated Sexagesimal System................................................................... 76 04.7. Peculiarities of the Mesopotamian System................................................ 78 04. 8. In Spite of the Shortcomings … ................................................................. 83 04.9. Why Was Base Sixty Used? ........................................................................ 85 04.10. Why Were Hybrid “Baby-Numerals” Used? ......................................... 90

05. Detour in Zero's History...................................................................... 92 05.1. Introduction ................................................................................................... 92 05.2. Two Current Usages of the Symbol “0” .................................................... 94 05.3. Why the Absence of “0”? ............................................................................. 97 05.4. Post 700 BC .................................................................................................. 102 05.5. Ancient Greek Role: “Greek-Number-Gate” ......................................... 105 05.6. The Role of the Ancient Indians ............................................................... 116 05.7. The Arabic-Islamic Civilization Role....................................................... 122

vi 05.8. Transmission of Zero to Medieval Europe ............................................. 125 05.9. The Mayan Civilization............................................................................. 128 05.10. Thou Shalt Not Divide by Zero! ............................................................. 129 05.11. Zero is Still a Mischievous Number ..................................................... 132

06. Arithmetic.......................................................................................... 133 06.1. Arithmetical Tablets ................................................................................... 133 06.2. Mesopotamians Were Numerical Analysts ............................................. 135 06.3. Tables For Reciprocals of Numbers......................................................... 138 06.4. Additions and Subtraction......................................................................... 142 06.5. Multiplication and Division ....................................................................... 144 06.6. Squares and Cubes of Numbers................................................................ 146 06.7. Square Roots and Cube Roots .................................................................. 149

0.7 Algebra ............................................................................................. 155 07.1. Importance of Abstraction ........................................................................ 155 07.2. Solutions of Linear Equations................................................................... 161 07.3. Solution of Quadratic Equations .............................................................. 163 07.4. Alternative Method For Solution of Quadratic Equations................... 166 07.5. Solution of Cubic Equations......................................................................168 07.6. Solutions of Simultaneous Equations....................................................... 170 07.7. Elimination Method for Solving simultaneous Equations.................... 171 07.8. Quadratic Simultaneous Equations ......................................................... 172 07.9. Finite Sequences .......................................................................................... 174

08. Geometry ........................................................................................... 178 08.1. Introduction .................................................................................................178 08.2. What were Ziggurats for? ......................................................................... 182

vii 08.3. Need of Geometric knowledge................................................................... 183 08.4. Knowledge of Pythagorean Theorem....................................................... 188 08.5. Yale Tablet ................................................................................................... 190 08.8. Susa Tablet ................................................................................................... 192

09. Geometric

Arithmetic...................................................................... 195

09.1. Introduction .................................................................................................195 09.2. What is a Pythagorean Triple? .................................................................196 09.3. Importance of Pythagorean Triples ......................................................... 198 09.4. How to Generate Primitive Pythagorean Triples? ................................ 202

10. To Prove or Not to Prove. That is the Problem.................................. 208 10.1. Introduction .................................................................................................209 10.2. The Greek Notion of Proof ........................................................................ 210 10.3. Too Magnificent Structures For the Real World................................... 211 10.4. Euro-Centric View of the History of Mathematics................................ 213 10.5. Example: Primitive-Pythagorean-Triples Theorem.............................. 214 10.6. example: A Proof of the Pythagorean Theorem..................................... 216 10.7. Example: Approximation methods .......................................................... 223 10.8. Example: Solving Equations ..................................................................... 225 10.9. Example: Sequences and Pie(s).................................................................228 10.10. Why Didn’t They ‘Publish’ Their Proofs?............................................ 229 10.11. A Fictitious Story ...................................................................................... 231 10.12. Conclusion.................................................................................................. 233

11. Astronomy ......................................................................................... 235 11.1. Introduction .................................................................................................235 11.2. Two Famous Mesopotamian Astronomers ............................................. 237

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viii 11.3. What Was Recorded in Those Diaries?................................................... 238 11.4. Mutual Relations: Astronomy with Mathematics.................................. 239 11.5. Predictions of What the Moon Does......................................................... 240 11.6. Predictions of Eclipses................................................................................ 242 11.6. Construction of Calendars......................................................................... 244 11.7. Mythologies Behind the Celestial Observations..................................... 245

Appendix (Mathematics) ......................................................................... 247 App.1. Introduction.............................................................................................. 247 App.2. Mathematical Foundations of Place-Value System............................ 248 App.3. Binary System .......................................................................................... 250 App.4. Duodecimal System ............................................................................... 254 App.5. Proof of the Pythagorean-Triple Theorem ......................................... 259

Appendix (History) .................................................................................. 265 References ............................................................................................... 269 Index ....................................................................................................... 272

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Acknowledgments To my professor an d great friend

Dr. M. W assel Al -

Dh ahir for initiat in g me To lo ve wh at I do. To my won derful, bett er h alf Nouria

A . A wakim for

patien tly lettin g me Do w hat I lo ve. To my belo ved t wo son s N abeel an d W aseem Sheko ury fo r intro ducin g

me To the world o f com puters.

Preface Those who cannot remember the past are condemned to repeat it. – George Santayana

We believe it is good to rediscover the childhood sense of wondering and marveling about some familiar daily realities, such as the following: • The seven days in a week. • The twelve months in a year. • The twenty-four hours in a day. • The sixty minutes in an hour, sixty seconds in a minute. • The 360 degrees division of the circle. • The place-value system of numeration, which enables us to express any number in a compact way. Some readers might take these daily realties for granted, and they occupy little “space” in their thinking. In fact, most people do not know, or even do not care, when these familiar things were established and by whom. If somebody cares for a quick and brief answer then, we say that some ten thousand years ago, many ancient people, among whom Mesopotamians were pioneers, had established these things. Our modern societies have inherited an important legacy from those ancient civilizations. Even our languages have traces of that legacy. Does the reader know that the word “dozen,” used in so many modern languages, is a corruption of “durzan”, which is a Mesopotamian word of an early epoch? Yes it is. It means “one fifth”; it is one fifth of their base number sixty. The present situation of the world, after the terrorist attacks of September 11, 2001, and the invasion of Iraq, calls for a consideration of history from a broader perspective. The West should be reminded that much of its present culture had several tributaries from different parts of the world such as Ancient China, Ancient Egypt, Ancient India, Mesopotamia, and others. Those ancient civilizations had contributed a great deal to what is called the “Greek Miracle.” Poincare` (1854-1912), the great French mathematician-philosopher said: “The greatest miracle is the fact that miracles never happen.”

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There were no miracles, Greek or otherwise. In fact, many advanced ideas and systems passed to the Greeks from earlier civilizations, which had developed concepts that are now wrongly credited to Western sources. In these turbulent days, a reexamination of the contributions of the Arabic-Islamic civilization is also necessary for a full understanding of the origin of an appreciable amount of scientific, mathematical, and philosophical concepts. This book is a modest attempt in the direction of reorientation of that perspective. The book is doubly restricted. On the one hand, it is mainly limited to the Mesopotamian civilizations, and on the other hand, it deals only with the fields of mathematics and related subjects. The style of the book is not a pure historical exposition, but an entertaining, leisurely “nonlinear” presentation with many speculations and diversions here and there accompanied with witty and “spicy” remarks. Several fictional stories meant to illustrate ideas presented or to bring home conjectures made are typed indifferent font. The study of the diverse contributions of the Mesopotamian civilizations is a huge undertaking that requires a lifetime occupation. A satisfactorily complete treatment of all aspects of Mesopotamian accomplishments would be exhausting and beyond the capabilities of a single author. I, being a mathematician, cannot but restrict my endeavors to the fields of mathematics and to a certain extent to astronomy. One of the goals of this book is to show how one culture, even if limited to one particular field, may enrich the progress of advancement of knowledge, and to show that, at least, the history of mathematics is far more global than had been previously thought. Many mathematical concepts originated in civilizations outside the Greek sphere. The view that this book tries to offer is that of an eagle soaring up in the skies perceiving the mountains, hilltops, and valleys, discarding the terrestrial minute details. It scans in a global manner the main mathematical contributions of the people who lived in and around the Mesopotamian valley from the earliest historical times until about 500 BC when that ancient civilization collapsed. Mesopotamians did not build their civilizations from scratch. No civilization did. Civilizations cannot be established in a cultural vacuum. The early people must have learned and inherited many things from still earlier people. Upon that inheritance, they were able to build further achievements. CHAPTER 01 is devoted to the general description of those primitive human beings, who evolved their mental capabilities throughout thousands of years of their existence, to reach the stage of possessing the primitive knowledge of, say, counting of a few numbers.

iv Some of those who reached that stage of mental evolution inhabited parts of the land of Mesopotamia. CHAPTER 02 briefly introduces the reader to the geography of this land and to a very brief history of those ancient people who throughout three or four thousands of years had established advanced civilizations. It is true that those people were, at different epochs, ethnically different, but astonishingly they stand possessing more or less a single culture, which can be described by one collective adjective, “Mesopotamian.” As modern society has CD’s, so the Mesopotamians had “CD’s too, which were the Cuneiform Dug-Oven-baked-Clay-Tablets that were excavated from different parts of the country. Those ancient CD’s speak eloquently through the cuneiform writings about that advanced culture. CHAPTER 03 is devoted to the significance of those tablets and to the importance of the writings inscribed on them. One of the Mesopotamian inventions is the notion of place-value system of numeration. It is the system in which numbers are expressible by utilizing only a manageably few symbols. This invention is considered a great mathematical development. It is true that Egyptian, Chinese, Indian, and Mayan civilizations developed more or less an analogous system, very likely, in an independent manner. Nevertheless, it seems that the Mesopotamians preceded them all. CHAPTER 04 presents this mathematical creation together with the peculiarities and defects of the Mesopotamian version. One of the defects is the lack of an explicit symbol for zero. CHAPTER 05 provides a “space-time” detour in the history of this “mischievous” number. A conjecture will be presented to explain the reasons for that lack, which persisted for thousands of years. In that detour, we will take-off via this chapter from Mesopotamia at the seventh century BC (that is only two centuries before the collapse of the Mesopotamian Empire). At that time zero was conceived for the first time in that land. The embryo was likely transferred to the Indian soil. The Chapter describes the labor of the birth of zero in that Indian environment. Zero, after being born, began to hike around, sometimes it got lost. The Chapter portrays short itineraries of the zigzag wandering of zero around the world; finally, at about the 15th century AD, it was accepted as a legal respected citizen in the “republic” of numbers. We will perceive, as we travel many ancient lands possessing great civilizations, such as the Egyptians, Indians, Greeks, and we will learn about the “Number-gate” that inhibited great Greek mathematicians from developing arithmetic and algebra. On the “flying carpet”, we will read from Rubaiyat of Omar AL-Khayyam1 a slight paraphrasing of the famous verses:

Incidentally, Omar Al-Khayyam was a first class mathematician and astronomer besides being a great poet.

“A book of verses underneath the Bough, A jug of wine, a loaf of bread “Zero” beside us singing in the Wilderness of the sky. Oh, Wilderness were Paradise enowl.

We will have a quick overlook of the Arabic-Islamic contributions to knowledge in general, and mathematics in particular. As we travel, we will have a glimpse, only a glimpse, of a far away land on the opposite side of the world, which was still unknown and unnamed. The land was populated with people establishing an independent great civilization. Today, that land is called Central America; and those people are known today as the Mayan CHAPTER 06 turns us back chronologically, as well as geographically, to Mesopotamia. We observe how the people over there, were manipulating their arithmetical operations with sixty as the base for their place-value system. CHAPTER 07 explains the abstractions of arithmetic into algebra. Abstraction eventually had to take place as long as there were mathematical minds around. In this Chapter, we make a quick digression into the role of abstraction that increases the applicability of mathematics. The Chapter describes how the Mesopotamians were able to solve first, second, and even some third degree equations as well as some simultaneous equations and how they dealt with some finite sequences. The construction of temples, building of high ziggurats, digging of canals, and erecting of embankments need not only numbers and their manipulations but also geometry and trigonometry. CHAPTERS 08 and 09 are allocated to the consideration of those two subjects. The latter Chapter is reserved to the most mathematically sophisticated tablet yet excavated, which is known as Plimpton 322. It contains a partial table of Pythagorean triples. The discovery of the algorithm for generating Pythagorean triples is usually credited to Diophantus (about 200 AD - about 284), a Greek mathematician who lived in Egypt’s Alexandria. However, some unknown and unnamed Mesopotamian mathematician had preceded Diophantus by about two thousand years in obtaining the result of the theorem and very likely its proof, too. CHAPTER 10 plunges us into a debatable topic. “To Prove or Not to Prove. This is the Problem.” Had those ancient people of Mesopotamia actually proved the results they had discovered? This is the problem. In spite of the fact that until the present day no tablet was found, as far as I know, that exhibits a proof of some of their results, yet there are some indications leading to a

vi conjectural belief that Mesopotamians did prove at least some of their results. We reached the conclusion: It seems that the Mesopotamians did not lack the notion of proof but they lacked a mathematician-librarian (like the eminent Euclid) who would have extended his main occupation of arranging manuscripts into arranging the known mathematical proofs and facts in a linear order. To arrange the known mathematical facts, he has to have some undefined terms and some initial statements to start with. We speculate that if the Mesopotamians had such a librarian-mathematician, they would have hit on the goldmine of axiomatic systems. If that is so, then, why did they not “publish” their results? No one knows for sure, they might have done that. Not all their tablets have reached us. Nevertheless, most likely, they did not. Why didn’t they? We think that the Mesopotamians, being a pragmatic sort of people like most engineers in our modern society, believed in and acted in accordance with the famous saying that: “The proof of the pudding is the eating.” They did not find it necessary to record the proofs of their results, as long as those results provide evidence of their usefulness in practice. A new perspective overall problem makes it irrelevant. CHAPTER 11, which is the last chapter in the book, presents a brief exposition of the wonderful achievements of the Mesopotamians in astronomy and the vast records, which they had kept, of great many observations about the motions of heavenly bodies (of course, without any telescopic instruments.) They used mathematical tools to measure the motion of celestial bodies. This had a feedback of astronomy playing a major role in the development of mathematics itself. Two appendices: APPENDIX (MATHEMATICS) and APPENDIX (HISTORY) are included: The first mentioned has five parts. A partial purpose of this appendix is to convince many of the readers of the existence of place-value numeration systems other than the familiar decimal one, which is usually taken for granted. The binary and the duodecimal systems are chosen to illustrate the ideas. The appendix also includes a proof of the classification theorem of primitive Pythagorean triples. The proof is straightforward and elementary in the sense that does not utilize any Mathematical WMD, that is, “mathematical weapons of mass deduction.” However, it is somewhat lengthy. APPENDIX (HISTORY) is a “time-line” about the rise and fall of Mesopotamian empires, wars, conflicts, conquest, and defeats. The mathematics in the book is elementary enough that any high school graduate can follow. However, it is of unequal level of sophistication. At some places, the reader may need a little refreshing of memory. Nonetheless, any reader who has forgotten his or her high school mathematics, and has some math-phobia, can skip the mathematical formulae and few mathematical footnotes with only a little loss in the “story.”

vii I hope that I have succeeded in convincing western readers that the Mesopotamian achievements were splendid and had helped the progressive march of science and that Western societies are indebted, at least in mathematics, to Mesopotamian and other eastern civilizations. Finally, yet importantly, I have to make a disclaimer. Focusing and highlighting on the achievements of the Mesopotamians should never be considered as an attempt on my part to dim the luster of contributions of other great ancient nations. Those had also great contributions, which even surpassed at some particular points that of the Mesopotamians. Some of those points are mentioned in the book. I owe a great deal more than the customary nod of appreciation to: Professor Melvyn Jeter, Chair of the Department of Mathematics and Computer Science at Illinois Wesleyan University for suggesting to me to deliver a seminar about the Mesopotamian mathematical achievements. Professor Jamal Nassar, Chair of the Department of Politics and Government at Illinois State University, for urging me to write this book, and for suggesting constructive criticisms of the manuscript. Dr. Aida Ovannesian for opening the Pandoraâ&#x20AC;&#x2122;s box of the computer. Dr. Thair AlBayati, Farouk Gewarges, Mehasin Jamma and the late Zuhair Hermes, for inviting me to deliver lectures at the Chaldian Club in San Diego. Those lectures eventually materialized into this book. It would take a better writer than I to convey the depth of my gratitude to Professor M. W. Al-Dhahir, from whom I learned mathematics a long time ago. Now he is giving me early encouragements. He read critically the drafts of the book, thoroughly edited it and suggested improvements. To him I am genuinely indebted. To my wife Nouria Awakim, I extend my thanks for the many joys she had contributed to my life. Moreover, she has a more direct responsibility for this bookâ&#x20AC;&#x2122;s existence. She has read the manuscript at various stages and suggested corrections and clarifications. I am fortunate to have her as my eternal companion. R.N.S. April 2005 Bloomington, Illinois rshekoury@mac.com

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MESOPOTAMIANS: PIONEERS OF MATHEMATICS

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01.

Prehistoric People

The most useful and least advanced of all human knowledge seems to me to be that of man […]. For how can the source of inequality among men be known unless one begins by knowing men themselves?

–

Jean-Jacques Rousseau

01.1. INTRODUCTION Though the general background for the rise of civilizations is well known, it is worthwhile to refresh the readers’ memories before elaborating, in later chapters, into

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topics concerning mathematical achievements of the Mesopotamian civilizations. All individuals of the human race, at prehistoric or historic stages, ancient or modern eras, are partners and collaborators in our endeavors to make Planet Earth a better place to live. It is a known fact that no civilization emerged from thin air or was established from scratch. Each civilization must have inherited or learned something from earlier people. Furthermore, the populace of emerging civilization must possess the capabilities and inclinations to adapt themselves to new conditions. There were many tributaries to the ancient Greek civilization, among which are the ancient Egyptian, the Mesopotamian, the Indian, and the Chinese civilizations. Those four civilizations and the Mayan civilization must have inherited and learned many things from prehistoric human beings. 01.2. THE EARLY STONE-AGE Let us have a quick look at the situations of Homo sapiens during prehistoric epochs between 38,000 and 8,000 BC. This was the era of the Early StoneAge, during which ice covered many parts of the Planet Earth. The earliest group of those living during this era did not know how to produce anything. Nature was cruel to them.

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They were consumers collecting their food from edible plants. Their living conditions were hard. They did not have permanent dwellings; they made temporary homes in caves or tents made up of tree branches, and lived in groups of not more than a hundred individuals. In short, the differences between the naked apes and the hairy apes were not very drastic. However, at a later stage of that era, the naked apes were able to make some rudimentary weapons that helped them in hunting and fishing. Males usually hunted and females gathered things. Frequently they had some kind of rituals upon successful hunts. Later, they learned to harness fire, which they used for cooking, for warming the places where they slept in, for lighting, for protection from wild animals, and for making torches to drive animals off. Certainly, their understanding of numerical concepts and space relations was very limited; it was probably slightly better than other upper apes. At a later stage of that era, some individuals must have had leisure time to enrich their lives by creative arts, featuring animals, Venus Figurines, and shamanistic drawings on the walls of their caves. A prehistoric â&#x20AC;&#x153;Leonardo Da Vinciâ&#x20AC;? must have made the following painting:

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Fig 01.1. The Cave Painting is from Lascaux, France about 20 000 BC.

There were some “Picassos” too!

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Figure 01.2

Drawing of Dancing Shamans. In France. Dated About 20 000 BC.

01.3. THE MIDDLE STONE-AGE The Middle Stone-Age began around 10,000 BC. Mother Nature played an important

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role in bringing the new Age. The ice sheet that covered Asia and Europe began to recede and eventually melted away. Fertile soil was exposed, ready to be agriculturally utilized. The Great Agricultural Revolution first took place where geographical conditions were favorable. It did not occur in the North, where winters were severely cold. It did not occur in the tropical regions where food was abundant, heavy clothing was unnecessary and therefore, there was no drastic need for improvement. But it occurred in regions where the conditions were not that so unhelpful that there is no chance compel change, yet not so adverse as to prevent attempts of farming and livestock raisings. Thus, the â&#x20AC;&#x153;human animalâ&#x20AC;&#x2122; of the Middle Stone-age gradually discovered or learned about elementary agriculture, which in turn caused slow but sure and permanent changes in the life style of those people. Their nomadic wandering in search for food gradually ended. Hunting and fishing were partially replaced by agriculture. Probably, the biblical legend of Cain killing his brother Abel was some sort of a collective reflection of this revolutionary transition from hunting to farming. The transition from gathering of food to its actual production was an immense revolution in the mode of life of early human

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beings. They learned that grains of corns would grow into new grains. That transition was a transformation, which turned manâ&#x20AC;&#x2122;s passive attitude toward Nature into an active one. There appeared attempts to control environment in order to regulate or enhance the production of food. In other words, the human being entered a stage of exerting some control on Nature. Consequently, this stage led to the necessity of understanding Nature.

01.4. EMERGENCE OF SETTLEMENTS There appeared at the banks of the great rivers in China, Egypt, India, and Mesopotamia, settlements. This occurred at about the same time (plus or minus a mere thousand years.) The inhabitants of those settlements tended to stay in one place as long as the soil of the locality remained fertile. As a result, some of those ancient human beings felt the need to build permanent dwellings near the riverbanks. Some did. Their dwellings were no longer caves but huts or tents covered with branches of trees. At a later era, they built granaries, baked bread, and even brewed beer. Moreover, they exerted control over wild herds. Consequently, villages began to emerge. Naturally, there appeared some sort of communications and interactions between neighboring villages, which can be described

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as a rudimentary commerce. These commercial interactions stimulated two opposing trends: Cultural and economic exchange on one hand, and the possibility of conflict on the other hand. Those conflicting trends tend to encourage making rules and agreements to avoid conflict. This is seen as an important factor in initiating the process of building up civilization. The above description of development of village communities in large parts of the Middle East seems to be the rule with only few exceptions. Necessity is the mother of invention, as the famous saying goes. Thus, the requirements of traveling between villages brought the invention of the wheel and the improvement on making boats. Agriculture brought the invention of the plow and the domestication of animals, especially the oxen, dogs, and donkeys. These animal species were â&#x20AC;&#x153;wiseâ&#x20AC;? enough to hitch their wagon to a potentially shooting star!

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Figure 01.3.

Invention of the Wheel

Can the reader imagine the world without the wheel? There would not be any car, or airplane, not even a carriage. There would not be traffic lights, no tickets for speeding. There would not be any cities but only be scattered villages. There would not be any highway. Transportation from one village to another would be carried on the back of a donkey. There would not be inspections of luggage as there would not be any hijacking unless to steal the donkey.

These should refresh our memories of how much modern civilization owes to the

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inventors of the wheel, who most likely were the Mesopotamians. Those achievements took thousands of years to be accomplished. However, one thing is certain; the tempos of technological, cultural, and administrational improvements were enormously accelerated in comparison with the previous stage of the Early-Stone-Age Period. Thus, after hundreds or even thousands of years, those technological advances slowly and steadily spread out among neighboring villages. Finally, but not necessarily at the same time, they became firmly rooted in some localities along the fertile valleys of the great rivers that could yield abundant crops. Great rivers are not all bliss; they flood periodically devastating everything. The seasonal flooding of rivers was a problem that faced those ancient people. The seasonal floods have to be controlled. These were not easy problems. Those settlements that were able to control the floods were among the pioneers in unfolding of a city-based society held together by economic enterprises, which by definition, is the beginning of civilizations.

01.5. EMERGENCE OF ADMINISTRATION

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The tasks of flood management, predicting their occurrences, building irrigation work, maintaining them and apportioning the water required large-scale coordination and cooperation of activities among separated villages. These necessary extensive organizations and collaborations eventually led to the establishment of central organs of administrations, which in turn created, in the long run, an urban aristocracy headed by powerful leaders. This was the embryonic appearance of city-states and eventually of empires. Those rapid changes not only led to the development of administrations but also to the rise of religions, the concept of law, literature and crystallizations of abstract notions of numbers2 and spacial relations, among many other things. The administration, even at the local level of a single village, needed a script to codify management of the requirements of village dwelling. Hence, some kind of record keeping and writing has to be developed. Not all settlements were capable to arrive at that phase of “sophisticated” development. Most historians believe that among the earliest people who were able to attain that 2 The unqualified word ”number” that appears in this Chapter stands for positive integers: 1,2,3,… Etc.

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stage were the Ancient Egyptians and the Mesopotamians at roughly about the same time plus or minus a few hundred years. We have to leave the great ancient Egyptian civilization aside, in order to concentrate on our main concern of dealing with Mesopotamia, the land, and its people, which is the task of the coming chapter. We refer the interested reader in ancient Egyptian civilization to the extensive literature on the subject; for example; [B] in the References.

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0.2.

Mesopotamia (ns)

Mesopotamia is the cradle of civilization - Kramer

02.1. THE MESOPOTAMIA

GEOGRAPHY

The ancient Greeks called the region between the two great rivers: The Tigris and The Euphrates as “Mesopotamia.” It is a region nearly 700 miles long and between 20 to 250 miles wide. In fact, the two Greek words “Meso” and “Potamus” mean “between” and “rivers” respectively. Thus, “Mesopotamia”

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is the land between two rivers. This region is roughly, what constitutes today’s Iraq. It is east of Syria and southeast of Turkey and west of Iran, The sources of the two rivers are very close to each other in the mountains of Turkey. Downstream their paths diverge. Euphrates passes through the present day Syria then goes through into Iraq. The Tigris passes directly into Iraq. Just before the twin rivers reach the Gulf in the South they confluence together forming a wide river. The Arabs call the merged river as “Shatt-Al-Arab,” which means, “The River of the Arabs.” It is 200 kilometers in length. The southern end of the river constitutes part of the border between Iraq and Iran. Various territorial claims and disputes over navigation rights between the two countries were among the main factors of the lengthy Iraq-Iran war (1980 – 1988). The Northern part of the country encompasses the foothills of the mountains of east of what is today’s Turkey. The Kurds of Iraq mostly inhabit those mountainous regions in the north. The present capital city is Baghdad, which is a large metropolitan city of about five million inhabitants. It is situated on the Tigris. Basra is the only seaport of Iraq on the Gulf. Mosul is the third largest city in the country. It is situated on the Tigris, too. Kirkuk is a

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moderately large city. It is an oil town. Irbeel and Suliamnia are the two main cities in the Kurdish region of the country. These are modern cities. They did not exist at the ancient era of Mesopotamia, with the possible exception of Mosul, which was the ancient Nineveh mentioned in the Bible. Among the famous ancient cities are: Babylon is situated on Euphrates near the modern Hilla city, Ur, Nippur, Lag ash, and Erode are now ruins in the south of the country. The present-day population of the country is around twenty-four million, four million of which have recently immigrated outside the country. The language spoken by the majority of the Iraqis is Arabic. However, the mother tongue of the Kurdish people is the Kurdish language, which belongs to the IndoEuropean family of languages. The Chaldians and the Assyrians, who are Christians, constitute about 4% of the population. . Currently many of them immigrated outside Iraq. Their mother tongues are dialects of the Aramaic language which is a Semitic language belonging to the ArabicHebrew family. Aramaic is the language that was spoken by Jesus Christ. Figure 02.1 is a map depicting the Ancient Mesopotamian region.

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Figure 02.1

A map of Ancient Mesopotamia and Modern Iraq

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02.2. THE CRADLE OF CIVILIZATION Some 12,000 years ago, the people that inhabited this region laid down the essential foundations of one of the earliest (if not the earliest) civilization in the world. Thus, many historians state: â&#x20AC;&#x153;Mesopotamia is the cradle of civilizationâ&#x20AC;? The earliest of the people who lived in Mesopotamia were probably the first to create a rich culture that would sustain itself and pass from people to people of other ethnic origin and pass from one religion to another religion and from one language to another language. The following are some of Mesopotamian pioneering achievements:

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• Inventing of writing. • Discovering of the place-value system of numeration. • Describing and recording of the complex motions of heavenly bodies. • Setting calendars synchronized with astronomical observations. • Building large temples decorated with artistic pottery and geometrically patterned. • Digging canals for irrigation. • Educating children at schools. • Developing of trade. • Inventing the use of money.. • Writing mythologies, and authoring immortal literature. • Inventing musical instruments: the flute, the drum, the trumpet, and the harp • Inventing the bad habit of gambling!

We realize that a full understanding of the scientific contributions of any civilization requires an overall comprehensive study of its political history. Yet, we will only embark, very briefly in the next section on the political history of the different Mesopotamian civilizations and will provide in APPENDIX (HISTORY) at the end of the book, a “time-line” of global picture of what had happened in the land of Mesopotamia. A more extensive treatment of political history would likely distract and divert the reader from the main goal of this work, which is restricted as mentioned earlier - to the mathematical contributions of the people who inhabited Mesopotamia up until five centuries BC. However, any reader interested in a complete political history of the country, which

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includes the rise and fall of several empires, is referred to the treatises [Kr] and [R], mentioned in the Reference list.

02.3. A VERY BRIEF HISTORY OF THE MESOPOTAMIANS The region had seen in ancient times several politically different civilizations. The following are the most important: Sumerian (3100 - 2050 BC), Akkadian (2350 - 2200 BC) Babylonian (2000 - 1600 BC), Assyrian (1350 - 612 BC), Chaldian (612 - 539 BC)

During the period between 10,000 BC and 500 BC, the fertile land of Mesopotamia had attracted not only immigration, but also several large-scale invasions, some of which resulted in establishing new civilizations. However, the new civilizations might have been politically and ethnically different, but they eventually became culturally almost the same as the former one.

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The Mesopotamians acculturated the invaders as well as the immigrants. After 539 BC, the Mesopotamian civilization collapsed. The Persian King Cyrus destroyed the centers of culture and the main cities. The Mesopotamian Empire vanished as a world power. However, its culture persisted for some centuries. It helped to build up and illuminate the Arabic- Islamic civilization. After the collapse of the last Mesopotamian Empire, the land began to be tossed around by occupying forces of this and that neighboring countries. Sorry! Not only neighboring countries, participated in occupying the land, but also countries that are away took part in the occupation. The British occupied Iraq at the beginning of the twentieth century, but also even the far away country of the United States occupied the land on March 18, 2003. The Americans occupied the land of Iraq in order to bring democracy into the Middle East.

02.4. DIFFERENT CIVILIZATIONS IN MESOPOTAMIA

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Thus, the region had seen in ancient times several politically different civilizations. The totalities of the people, who inhabited and governed Mesopotamia, were ethnically diverse. Some were of Semitic origins, speaking languages of the ArabicHebrew family with slightly different accents. Others were speaking languages of the IndoEuropean family. The City-States were constantly at war with each other until more powerful people swallowed them. Some of the people were pacifists; others were brave warriors who were able to conquer regions as far as Egypt.

02.5. GROUPING THE MESOPOTAMIAN CIVILIZATIONS UNDER A SINGLE NAME The history of Mesopotamia is complicated. It takes volumes to adequately cover all of its ramifications. Nevertheless, no further historical details over the previous section will be given in this book. In fact, we will do the opposite. We will group together all of those civilizations under one term: â&#x20AC;&#x153;Mesopotamian civilizations.â&#x20AC;?

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Consequently, the Sumerians, Akkadians, Babylonians, Assyrians, Chaldians and others, will all be described in this work simply as “Mesopotamians”. We will not use other particular terms unless we have to. There are several reasons for employing this collective terminology: First: The purpose of this work, as was pointed out previously, is to give an eagle-eye point of view of the mathematical contributions of the people who lived in Mesopotamia until about five centuries before Christ, when the Persians occupied it. Therefore, any further historical detailed elaboration would defeat the purpose of this work. Second: Nowadays, even in spite of the passage of more than twenty five centuries, there are still significant sensitivities among some of the present day descendent of those who lived or used to live in today’s Iraq about who contributed what. Therefore, it is advisable to avoid bringing up or provoking this chauvinistic “allergy.” Third: The cultural “absorption and merging” of different civilizations makes the entry into priority research about the originality of a certain discovery or practice, difficult if not impossible.

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02.6. CULTURAL STURDINESS MESOPOTAMIAN CIVILIZATIONS

We think it is important to explain what is meant by â&#x20AC;&#x153;cultural absorption and mergingâ&#x20AC;?. The earliest people who lived in Mesopotamia were, as was pointed out earlier, probably the first to create a rich culture that would go from people to people, from religion to religion and from language to language. Throughout those thousands of years, there were constant migrations and several invasions of the land of Mesopotamia. Immigrants and invaders were culturally less advanced than the invaded Mesopotamians. Nevertheless, astonishingly for the great civilization that existed there, had a culture, which was constantly renovated by practical effects of urbanizations, writing and governmental administrations. Hence, the newcomers gradually found it beneficial for them to adopt the pre-existing culture (the government, economy, writing, mathematics, science, religion, laws, legends, and literature) of the invaded people of Mesopotamia and of course, adding a few things of their own here and there. Thus, after few generations the newcomers found themselves metamorphosed more or less into Mesopotamians! Thus, we observe that:

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Ethnically, people had changed. The spoken language had change. However, the culture including the method of writing language was preserved.

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03. Mesopotamian “CD’S” The Moving Finger Writes, and having writ, Moves on, nor all Thy piety nor wit Shall lure it back to cancel half a line Nor all the Tears wash a word of it.

– Omar Al Khayyam

03.1. INTRODUCTION The “CD” in the title stands for “Cuneiforms-Dug-Oven-Baked-Clay-Tablet.” There are some similarities between modern CD’s and Mesopotamian CD’s.

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Our modern compact discs are made of polycarbonate with one or more metal discs capable of storing information in digital forms. One needs a CD-reader in order to read these compact discs. Mesopotamians had their â&#x20AC;&#x153;compact discsâ&#x20AC;? too! Those were tablets made of baked clay. They were also capable of storing information in wedge shaped characters called cuneiforms. Approximately half a million Mesopotamian tablets had been excavated. The information that was stored in them ranges - just like modern CDâ&#x20AC;&#x2122;s - over a wide range of fields from Agriculture to Zoology. Exhibits of those tablets are found in great museums of London, Berlin, Istanbul, Leaden, Baghdad and others. Historians and archeologists assure us that those tablets contain, most likely, the first large-scale communication through writings. Almost everything that is known about ancient Mesopotamian civilizations came from deciphering those tablets. Some Ancient Greek tourists contributed further observational information too. About four hundred of those tablets contain mathematical problems or lists of mathematical tables.

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03.2. SOCIAL NEED FOR WRITINGS We discussed in the previous chapter the extent of the impact of the agricultural revolution that changed the life modes of human beings. Another important revolution followed. When villages attain a critical stage in capacity, deep changes occur in the fabric of social and political life of their inhabitants. This is called the urbanization revolution. Urbanization revolutions have profound influences. They greatly alter the inter-human relationship of the dwellers of the villages. Moreover, Urbanization and writing are closely related. In general, the emergence of cities at the beginning of an unfolding civilization creates enormous changes in inter-human associations concerning food production. The mere appearance of cities signifies that a large part of the population is tending to stop raising its own food. This tendency has a significant social and political impact. The part of the population, which raises its own food, has the responsibility of providing food for those who do not. Otherwise, urban society would collapse, and no urbanization would take place. We suspect many did collapse,

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Therefore, in order to keep the momentum of urbanization going, the king or the governmental administrations of the community or city-state should be able to face an important problem. It is the problem of efficient well-organized food distribution. Efficient and well-organized distribution, in its turn, necessitates the existence of a bureaucracy that carries the dealing, delivery, and record keeping. It is obvious that record keeping calls for writing in some form or another. If a society is not able to evolve into the stage of efficient record keeping, then it cannot blossom into a full-fledged city community. The early inhabitants of Mesopotamia were most probably quite successful in responding to the requirements of sustaining their society in its most critical stage, the embryonic stage. One of the things that sustained their rising society is their invention of writing. The prime motivation of early Mesopotamian writings was of an economic nature: the desire to administer economical and trade interactions. Almost all of the very early cuneiform texts and a very large fraction of the second millennium texts were concerned with economy and administration.

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In fact, many thousands of excavated stone tablets that were dated back to the period between 8,000 BC â&#x20AC;&#x201C; 3,000 BC, carried inscriptions of writings and numbers recording distributed goods. These should remind modern people of the receipts they get in the stores. Naturally, writing evolved throughout the thousands of years from one stage to another. Let us have a look at what kind of writings the early Mesopotamians had at the first stage.

03.3. FIRST STAGE OF WRITING From the earliest period, the clay tablets, tokens in various forms and shapes were used as counters. Each symbol represents some concrete thing, for example a bullâ&#x20AC;&#x2122;s head or a sheep or a basket. That is, Mesopotamians depicted a concrete object by a picture of that object. Such writings are called pictograms (that are drawings representing actual things). The tablet depicted in Figure 03.1 illustrates an actual pictogram Mesopotamian tablet. Since pictograms do not have pronunciations associated with them, they are

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often used even today at international airports and grand hotels because they refer to the object directly without the mediation of a spoken language. Almost everyone understands them. See Figure 03.2 for modern pictograms:

Figure 03.1

Pictogram Tablet

Moreover, â&#x20AC;&#x153;computer software iconsâ&#x20AC;? are nothing but pictograms. They are the fashion of the day!

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Figure 03.2 Examples of Modern Pictograms

However, by repeated use throughout long periods, the Mesopotamians were able to evolve their picture-words into shorthand wedged lines. The pictures began to look simpler, even abstract. These marks, in time, became wedge shaped. Eventually Mesopotamians were able to inscribe pictures that could convey sounds or abstract concepts. The first pictures were drawn in vertical columns. Then people began to write in horizontal rows from the right side to the left side as writing Arabic, Hebrew, Assyrian, and Chaldian are done today. Figure 03.3 shows the evolution of the words “ fish,” “ox,” and “duck.”

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Figure 3.3. The evolution of some words

These writings are known as cuneiform writings. The Latin word “cuneiform” means wedge-shaped. The writings were produced by the use of reeds as writing instruments on wet clay and by moving the end closest hand back and forth once. The wet clay would then dry then baked in a kiln into stone-hard tablets. The tablets, in spite of being clumsy, are not too bad for record keeping because it is hard to lose one’s records if they are big heavy tablets! Furthermore, tablets are more permanent and durable. When, after some thousands of years from today, all the presentday taxonomy books will go to dust or ashes, the Mesopotamian tablet carrying local

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taxonomical information, whose picture is shown in Figure 03.4, probably, will not perish.

03.4. MEDIA OF COMMUNICATION Cuneiforms were the ancient media of communication. Mesopotamians kept documents about every object they acquired. Every business transaction they carried had to be recorded.

Figure 03.4. List of the kinds of local fish

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Near the gates of cities, scripts would sell their services. Their hands would move fast on a lump of clay. Then the contracting parties would add their signatures by means of seals. The usual seal is engraved in a cylinder of stone or metal that could be rolled over wet clay. Tablets with cuneiform were used for every purpose, just as writings on papers are used today: They were, for example, used in: •

• • • • • • •

Personal corresponde nce Narratives Laws Prayers Incantations Dictionaries Medicine Mathematics

Cuneiform tablets were also extensively used in schools. School tablets were usually recycled unless permanence was required for some reason or another. In such cases, they could be baked in an oven. The cuneiform writings spread to neighboring countries such as Syria, Anatolia, Armenia, Persia, and were used for a few centuries even after the collapse of the

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Mesopotamian Empire. Culture is more durable than empires.

03.5. SECOND STAGE OF WRITINGS The second stage of recording activities began to unfold. Scribes began to use a more complex system of notations in which symbols for numbers started to appear. They no longer were using the symbol of an ox five times in order to represent five oxen, but rather, they began to write down the ox pictographic symbol along side the symbol denoting the number five. This means that the abstract notion of number began to be crystallized. We will elaborate further on this evolutionary development of the notion of numbers in the next chapter. Simultaneously, other written symbols were developed on phonetic basis rather than on pictographic basis. This allowed the recording of more abstract items such as the names of gods, kings and humans. With this breakthrough, the recording of written language developed so that cuneiform writing not only registered or counted things, but could also tell stories and legends like the famous Epoch of Gilgamesh. This was a small step towards phonetic alphabets.

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It is regrettable that Mesopotamian writings never developed into a phonetic alphabet. Some can even venture to say that had the Mesopotamian civilization survived a little longer, it would have developed or adopted the phonetic stage. Speculation aside, humankind has to wait until about 900 BC for other people with a weaker attachment to their legacy of writing. That is, we have to wait for the Phoenicians (who were mentioned in the Bible as the Canaanites), who incidentally were likely to have been influenced by the Mesopotamians during the many military conflicts. For a glance at the historical events, see APPENDIX (HISTORY)

03.6. LIBRARIES Mesopotamians did not only write tablets to be discarded away, but they had a deep sense for collecting them in libraries. The most famous libraries were in Nineveh, the capital city of one of the great civilizations of Mesopotamia, situated near the present day Mosul in the north of Iraq. The city had great canals that provided water from the Tigris to the municipal gardens that were stocked with unusual plants and animals.

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Some of the ruins of the city had been recently restored. We are interested in the libraries of Nineveh. The two palaces of King Sennacherib and his grandson Assurbanipal contained archives of cuneiform tablets. Moreover, the library of Assurbanipal forms an unrivaled epigraphic source for current knowledge of Mesopotamian history. It was one of the greatest treasures of ancient Mesopotamia containing more than 20,000 tablets and fragments, many of which are copies of ancient Mesopotamian texts such as the famous Sumerian Epic of Gilgamesh and the Babylonian Flood story; their subjects range over literature, religion, sciences, mathematics, and dictionaries.

03.7. USE OF CUNEIFORM WRITINGS IN SCHOOL-EDUCATION

At the early era of Mesopotamian civilization, children, both boys and girls of the age eight or nine, were sent to the temple schools. Thus, schooling was initially associated with priesthood and took place in temples. Nevertheless, at a later age, this was changed. Education, apart from the temple,

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arose for youngsters of wealthy families who paid for the education of their children. Students were obliged to work hard from sunrise to sunset. Using a clay tablet as a textbook the teacher wrote on the left-hand side and the student copied the model on the right-hand side. The students were obliged to work hard at their studies and were encouraged with praises and applauses while their inadequacies and failures were punished with lashes from a stick or a cane! There is an excavated tablet, which can be considered as a report of a teacher to one of his students. The report says: â&#x20AC;&#x153;You have been here for two decades and have not learned a thing. In two days I will give a test. If you do not pass, I will beat you severely and throw you out to begâ&#x20AC;?!

This scolding report shows the extent to which the Mesopotamians esteemed learning. If a student did not learn, he had to beg in the streets! The students learned not only the cuneiform writings, but also arithmetic, the long table of multiplication in sexagesimal numbers, the classical Sumerian language, and its literature (though it was not a spoken at the time). Parenthetically, Sumerian tongue was the language of the learned and educated people, just as Latin and Greek were, a few years ago, in western societies.

- 39 -

However, the teaching was predominantly of practical nature and heavily relied on rote learning of poetry and complex grammars of the current spoken language and of Sumerian language.

03.8. DECIPHERING OF THE CUNEIFORM WRITINGS The cuneiform writings - as was pointed out in a previous section - spread out through the means of commerce and military conquests to many regions neighboring Mesopotamia such as Persia, Armenia and Anatolia. However, about a thousand year after the collapse of the Mesopotamian Empire in 539 BC, knowledge of cuneiform was completely lost. One factor for the loss of the language was probably due to the Phoenician development, at about 900 BC, of the much simpler phonetic writing, which very soon replaced the cuneiform writings. The reader is likely to be wandering, by now, how and when cuneiform writings were deciphered. The story is worth telling: Remember that the Egyptian Hieroglyphic writings were deciphered in

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1828, during Napoleon occupation of Egypt. With the encouragement of the famous mathematician Fourier (1768-1813), his former student Champollion (1790-1832) accomplished the deciphering. Sir Henry Rawlinson (1810 - 1895), a British army officer, who was initially appointed to help in reorganizing the Persian army and later was assigned to several political duties in the Middle East. He was appointed in 1853 as the British Consul in Baghdad. Later, he was assigned as the British Ambassador to Persia. Rawlinson spent so much time in the Middle East, that he learned modern Persian and other Oriental languages Sir Henry is best known, not for his skill in diplomacy or military organizing, but for his decipherment of ancient cuneiform. The deciphering of the Egyptian Hieroglyphic writings in 1828 encouraged him. Rawlinson made some initial attempts to decipher the strange shapes in the excavated tablets. In Persia, there were many cuneiform tablets with visible inscriptions. However, most of them were very short, consisting only of a few characters, which would not help much in deciphering. However, a considerable lengthy inscription was known to exist in the mountains of Hamadan in Persia. High up on a mountain called Behistun (see Figure 03.5)

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there was a panel of sculptured figures with many lines of cuneiform in three different scripts. Those writings also appeared on several other Persian artifacts. The writings were carved during the reign of King Darius of Persia (522-486 BC). They consisted of identical texts in three languages written in the script of Old Persian, Elamite and of Mesopotamians cuneiform. Between the years 1835 and 1839, Rawlinson, with great risk to his life, succeeded in copying down most of the great Behisitun inscriptions. He began his deciphering research by supposing that the three different types of writings read the same thing. One of the three was simpler than the others. Its characters were less complicated in form and fewer in number It appeared to be alphabetic while the others seemed to be pictogram, characters. Rawlinson began with the simpler of the two, which was found more often throughout Persia and had a hint of Persian dialect. Darius the Great of Persia constructed the Behistun inscription about 519 BC. It gave an account of how he came to the throne after the death of Cambyses and how Darius overcame those who threatened to destroy the unity of the Persian Empire. This statement of Darius was widely known throughout his realm. One copy of the

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inscription was found written on a papyrus in southern Egypt in the Aramaic language and alphabet. Once the Persian text had been translated, it was possible to turn to the study of the other two texts. One was correctly assumed Babylonian. This discovery proved to be very important to students of Assyriology since Babylonian and Assyrian languages were both Semitic and closely related. It should be reminded that before the 1830â&#x20AC;&#x2122;s there was almost a complete ignorance about ancient Mesopotamian history. It was soon learned that the cuneiform system had been used by many different groups and for writings in a variety of languages. Semitic speaking Babylonians and Assyrians used the cuneiform writings for thousands of years, but later archeological discoveries showed that the Sumerians were the inventors of the cuneiform writings before 3000 BC.

- 43 -

Figure 03.5.

Behistun Inscription

After translating the Persian, Rawlinson was able to decipher the others. By 1851, he could read 200 cuneiform signs. Rawlinsonâ&#x20AC;&#x2122;s work resulted in a breakthrough of many other discoveries

- 44 -

concerning Mesopotamia. The deciphering provided a great insight into human history.

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04. Numbers, N

umerals

Number-Systems One who cannot count does not count. –

Anonymous

04.1. INTRODUCTION

The three capitalized “N” ’s appearing in the title of the chapter require some clarifications. Numbers are concepts. They exist in the minds of human beings. In order to talk or read about concepts one needs written or spoken conventional symbols that represent them. Numerals are symbols for numbers. Numerals have material existence. The following examples give further clarifications. Justice, Beauty, and Love are concepts. They also exist in the minds of human beings.

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Figure 04.1. Justice, Beauty, and Love The above figures are conventional symbols that represent them. They are material things either statues or pictures. Statues of Justice and Venus are stones. A picture of Cupid is a painting on Canvas.

- 47 -

Now, back to the concepts of numbers! The number three, for example, is an abstract concept that exists in the minds of

“3”

people. However is a conventional symbol that represents the notion of the number three. It is a material thing. It is an ink spot! Of course, there may exist other conventional symbols for the same concept.

For example, the symbol also represents the number three. This particular representation is used in most Arabic countries. On the other hand, both “1+2” and “1+1” are representations of some numbers. The following 1+2 = 3 is a statement claiming that the symbols “1+2“ and “3” represent the same concept. This is a true statement according to the theory of arithmetic and the standard conventions. The statement, 1 + 1 = 3 claims again that the two symbols, separated by “=” represent the same concept. It is a false statement by the same mentioned standards.

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Now, after explaining the first two “N”s that appear in the title of the chapter, let us turn to the third “N”, namely, “Numeration-Systems.” Numbers do not end. In the precisemathematical language, we say, “The set of numbers is infinite.” Consequently, assigning a distinct symbol for each number requires the use of too many symbols. In fact, it requires an infinite collection of symbols. Of course, this is impossible to accomplish. Our life is too short for such a feat! Therefore, through thousand of years different societies developed schemes by which reasonable collections of finite symbols can generate representations for as many numbers as possible. These schemes are called “Numeration Systems.” Naturally, there existed through history many different numeration systems, most of which, at present, are considered obsolete. The Roman system, I, II, III, IV, V, … etc, that persists mainly for ornamental purposes, is inadequate for the present day scientific-technological society. The numeration system that is currently adopted all over the world of today is the decimal place-value numeration system. This system uses only the ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. This set, though finite, but adopting some “trick of convention”, can be made to potentially generate symbols for ALL numbers3. The trick 3

- 49 -

allows the same symbol to represent different numbers, depending on the position of the symbol. This is the reason for the system being described as a “place-value system”. We will give further clarifications in subsequent sections as well as in APPENDIX (MATHEMATICS). A long time ago the Mesopotamians, the ancient Egyptians, and Indians invented the place-value system, most likely, in an independent manner. The Mayan society also developed the same system definitely independent of the others. The Mesopotamians were, most probably, the pioneers in this respect. Moreover, the Mesopotamian system is unique. It is a radical departure from the rest of all other ancient numeration systems, since it used sixty as a base, while the other systems used ten or twenty as bases. Now, the three “N”s of title are explained. We turn, in the rest of the chapter, to the problems related to treatment of how the ancient human being, in general, and the Mesopotamian, in particular, invented or discovered the notions of numbers and numeration systems.

04.2. PROBLEMS OF THE EMERGENCE OF THE NOTIONS OF NUMBERS

The important word in the sentence is “potentially.”

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Modern society extensively uses, in everyday life, the concept of number as well as the word “number” in several ways. We use numbers as cardinal numbers, as ordinal numbers, and as nominal numbers 4. As a result, we tend to take the ideas of number for granted. However, an inquisitive mind raises several questions such as:

How did the ancient human b eing develop the abstract notion of numbers? • Did every culture develop the ideas of numbers separately? Or, have they arisen in only a few centers of cultures and then spread throughout the world by trade and war? • Is tallying the same as counting? •

A lot of academic research had been done about these problems. The topics 4

Ordinal numbers are used to indicate the order or rank of things in a set, for example: first paragraph, fourth runner, and tenth place. Cardinal numbers are also known as the “counting numbers” for indicating quantities. They are: 0,1,2,3… Nominal numbers are used to name or identify something, for example, a social security number, phone number, zip code. These do not represent quantity or rank. There is no meaning in being arithmetically manipulated.

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surrounding these queries are interlocked with each other and with other cultural factors related to writings, languages and technologies, â&#x20AC;Ś etc. Some readers may wonder why these questions are considered difficult problems for the ancient human being and had remained difficult for thousands of years. These questions seem to those readers irrelevant or even trivial. These problems seem to be easy because the doubting readers had learned about numbers at a very early stage of life. We were â&#x20AC;&#x153;nursedâ&#x20AC;? with numbers. In fact, modern societies are keen to start teaching their citizens about the concepts of numbers, numerals, number-words, numeration system and manipulations with them, at about the age of four. The complete teaching task is usually accomplished in about three or four years. However, it took several thousands of years for the ancient human beings to attain the same level. The ancient people struggled very hard to learn or to discover the concepts of numbers and related facts. The stages of learning about numbers are similar in both modern child and ancient human being. In order to perceive the similarities simply collapse every thousand years of

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ancient man into one year of teaching for the modern child.

The next two sections will deal with the twin-problems of the stages in the process of learning of a typical modern child and of a typical ancient human being about the abstract concepts of numbers, number-names, numeration, and related subjects.

04.3. INITIATING THE MODERN CHILD IN THE NOTION OF NUMBERS Let us refresh our memories and watch the stages of progress in the process of initiating a typical modern child in the notions of numbers, the representation of numbers and their manipulations. The initiation starts at the preschool or the kindergarten age and last for three or four years. One notices, upon observing the childâ&#x20AC;&#x2122;s textbooks, and watching teachers and parents, that the child is painstakingly drilled into the concepts of numbers through consecutive stages.

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The initial step of the child learning starts with the concept, which mathematicians call â&#x20AC;&#x153;one-to-one correspondence.â&#x20AC;? The child is made to practice - as a favorite playful activity - in drawing arrows from diagrams of a collection of objects into another collection. The objects of the collections are chosen from his surroundings or his favorite toys. For example: The child is shown a pair of pencils, a pair of toys, and a pair of pictures of spider man or batman. The child is presented with a trio of apples, a trio of trees, a trio of dinosaurs, and other collections of fours and five objects. The child is asked to join the corresponding elements of the collections by drawing arrows. These activities are repeated with increasing sophistications many times for several weeks or months. While the child is engaged in his favorite activity, the teacher or the parents, often repeatedly pronounces the corresponding number-word. Further corroboration of the indoctrination of the child in the notion of oneto-one correspondence is accomplished by encouraging him5 to use his own body. Thus, the child learns that his eyes, are in one-toone correspondence with his ears, and his 5

Or her.

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fingers in one hand is in one-to-one correspondence to the fingers in his other hand, … etc. The members of the child’s family or a collection of child’s friends are also utilized to drill in the concept of one-to-one correspondence. The child, through numerous repetitions of these activities and practices over a long period, gradually begins to perceive that there is something in common among the collections of pairs, and there is something in common among the collections of trios. Similarly, there is something common for the collections of fours and fives, … etc. Thus, the child, while still at the beginning stages of learning to read and to write, dissociates that particular common property, from the things in the collection. In other words, the child is about to mentally reach, in a rather foggy manner, to a few abstract concepts of “baby” 6 numbers, maybe: one, two, up to five or may be up to ten. At the same time, he is made to learn the names of

Putting the word “small” number between quotation marks to is indicate, in a short hand manner, that it is less than or equal to the base of the system. In fact, there are no small or large numbers. However, there is a less than relationship, for example 2 is less than 3. This does not make 2 small or 3 large.

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those numbers in his mother tongue and their standard conventional symbols. The child, at the same time, is trained into finger counting and the use of other parts of his body without drawing arrows. Thus, the ideas of one-to-one correspondence are corroborated and the newly learned numbers and their names are supported. The child, at this stage is likely to be fascinated by the names of “large numbers” such as “thousand” and “million.” A little later, the child is instructed, with the aid of diagrams and arrows that those “small” numbers can be arranged in a natural order. Consequently, he begins to realize that for every number there is a larger number. Hence, the child picks up the idea that there is no largest number; therefore, the sequence of numbers cannot end. For any “large” number such as a million, a million and one or even a million times a million that is larger than a million. In other words, the child will have a dim glimpse into what mathematicians call, an infinite set. The set of numbers is infinite. These stages of learning will take from two to four years. Hence, the modern child, at about the age seven or eight has a vague realization of the impossibility of giving each number an independent symbol or an independent name. Thanks to our great-great… - great ancestors who solved the problem more than four thousand years ago. The

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solution settled the problem and gradually spread it throughout the reluctant world. Modern educationists have inherited a “canned” solution; it is the decimal placevalue system. The educationists’ ingenuities lie only in choosing the most appropriate modern technological “can-opener” to present to the modern child. The favorite “canopeners” used to be abaci.

Your Majesty Abacus complete with a mouse

Figure 04.2

Abacus “Complete with a Mouse”!

As a result, the child will realize that, for example, the three 2’s in the number 235322 stand for different numbers. “The extreme left numeral 2” is read as two hundred thousand, the “middle numeral 2” is twenty and “the extreme right numeral 2” is simply two; and analogous statements can be said for the pair of 3’s one of them stands for thirty thousands, the other for three hundreds.

No ! Your Majesty After 6 months it’ll be half price

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The indoctrination is usually so efficient that many children will take the system for granted when they grow up. It takes great pains to convince them that there are other place-value systems as good as the decimal system. In fact, APPENDIX (MATHEMATICS) gives two examples of different place-value systems. In the final stages of initiation, the child will be drilled into the famous four operations of arithmetic, namely: addition, subtraction, multiplication, and division7. Therefore, we see that the whole process is far from being an intuitive straightforward method to learn about numbers. It takes the child several years with the assistance of sophisticated educationalpsychological technology, which is continuously renovated by thousands of research. Before closing this section, let us make a clarifying remark: We do not mean that the child has, at the different stages a crystal clear 7

However, alas, many children in advanced countries, because of the too early use of calculators, will forget about these manipulations. This might not seem so strange. Havenâ&#x20AC;&#x2122;t most of us forgot the art of kindling a fire by flints? Our great-great-grandparents were experts in this art.

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idea of what he is being taught. We only mean that the child would get ideas in a dim manner, but enough to proceed forward to the following stages of learning.

04.4. THE GRADUAL DISCOVERY BY THE ANCIENTS OF THE NOTION OF NUMBERS The records of early numerations that are discovered today seem to come from Mesopotamian and Ancient Egyptians only. Little is known, with a fair degree of certainty, concerning similar records in ancient China and India. Upon comparing the progress in learning about numbers, by ancient human beings with that of a modern child, we observe existence of similarities, as well as differences. To catch sight of the similarities, simply (as we have stated before) collapse every thousand years of ancient civilizations into roughly one year in the child progress. When compared, the ancient man with the modern child, the former was at a disadvantage The modern child lives in a culture that provides him or her with somebody to teach the abstract ideas of numbers, and related topics.

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While the ancient man lived in an almost a cultural vacuum:

No concepts. No number-name. No number symbols. And certainly no numeration system and no teacher! In almost all civilizations, rudimentary counting is nearly as old as speech. We mentioned in an earlier chapter that when people, in the early Mesopotamian civilization, (namely, from 8000 to 3500 BC) wanted to write “five oxen”, for example, they would draw a symbol for an ox five times on a clay tablet. These findings indicate that those ancient people, at that period, had not yet grasped the abstract notion of numbers, not even the “small” numbers. Numbers did not have independent existence. They were still “locked” with things; yes, they did see five oxen, but they were not aware of the “five” in the scene of five oxen. In other words, those people did not yet discover that there is something in common among “five oxen, “five trees”, “five apples”, and “five dogs.” In contradistinction to the modern child, those ancient people did not have a “teacher” to point out, to design a scheme to highlight, and bring out to their minds that common property in all these sets. The ancient people had to stumble on some things that ignite their minds (or more

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precisely the minds of some of them) that illuminate the concept of one-to-one correspondence in the difficult, and long way. Evidently, they did stumble. How did they stumble? Nobody knows for sure. We can only conjecture. Let us convey what might have happened as a fictitious story:

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At about 6000 BC there was an exceptionally intelligent man named Sargon living in Eridu, a village in the south of Mesopotamia. He was in the habit of taking his whole family, his wife and his six children, along with him for hunting rabbits and usually gave a rabbit to each member of the family. But taking the children with him proved to be a burden; he had to keep his eyes on them; otherwise, they might get lost around the swamps in the south of the country. He had never forgotten the daughter he lost in one of his hunting expeditions. On the other hand, keeping his eyes on the children distracted his attention from catching rabbits. It was difficult for him to do the two things simultaneously. Ever since losing his daughter his mind was occupied with this problem One day, on an early morning he woke up. The rest of the family was still fast asleep. As he was rubbing his sleepy eyes, and gazing at a particular tree, an idea flashed into his mind. It came from nowhere. Thinking aloud he said to himself “Why not get that dried tree branch over there, and make scratches on i t. A scratch for the wife, a scratch for this child, a scratch for that child, and so on up to the sixth child?” He continued still addressing himself: “Keep wife and children home, take the scratched piece of wood with you, catch a rabbit for this scratch, a rabbit for that scratch, so on. Bring the rabbits home. Distribute them to each member of the family.” “EUREKA!” he shouted so loud that awakened his wife. She asked what happened? He did not bother even to answer her. He was so immersed in his thoughts as if he were possessed by the devil. He reached for the branch of the tree, and the sharp stone. Glanced at his wife, made a scratch on the piece of wood, turned to each of the sleeping children made a scratch on the piece of wood for each of them. He darted without uttering a word to the bewildered and confused wife. She was very happy when very soon he returned home, after a while, carrying a bunch of rabbits. Each member of his family got his or her share was very much delighted.

Sargon scheme worked out. He won the admiration of the whole village.

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His achievement was an important scientific-technological breakthrough. It was the first glimpse into the mathematical notion of oneto-one correspondence. However, Sargon did not realize how deep and important his breakthrough was. He never imagined that this was a step to carry man to the moon. Something analogous to the fictitious story must have actually happened. However, let us continue with the story to observe the impact of the new discovery. Sargon passed his invention to his friends. After a few years, the whole village was using this technique for many varied purposes. Some of his friends made slight modifications that suited their particular purposes. Some used sticks with fewer scratches, some with more. Some made scratches on bones instead of dried branches of trees. Neighboring villagers, through trade, learned about the new invention, used it in their villages. The process was even given a name. Today, we call it â&#x20AC;&#x153;tallying.â&#x20AC;? The discovery, invention or whatever one wants to call it, was passed to future generations. After a few centuries, knowledge of this technique spread to the whole of Mesopotamia.

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Immigrants learned about it, as well as invaders. After the passage of some half a dozen centuries, the stick tallying became well established.

What happened later? Lot of people gradually began to realize that it did not matter what material the tally was made of, whether it was of wood, bone or stone. Moreover, also it did not matter how deep the scratches were and how they were distributed along the stick.

This was the beginning of a partial realization that: There is some common property among the tally sticks, which are in one-to-one correspondence with each other. The human beings at that time - as was pointed out earlier â&#x20AC;&#x201C; were living in a cultural vacuum. They were not, as fortunate as the modern child, who is privileged to have a teacher designing a scheme to accelerate the full emergence. The ancient human was destined to learn the hard way. Let us see whether there was any further improvement. At this stage of development, Mesopotamians discovered (in a vague manner) that there was something in common

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the tally sticks that are in one-to-onecorrespondence with each other. For example, there is something in common among “five oxen,” “five apples,” “five men,” and “five trees.” The common feature is the notion of “fiveness,” similarly about “oneness,” “twoness,” “threeness” etc. This perception must have been a conceptual leap in abstraction. This must have taken place at around 3500 BC. Nevertheless, after the passage of a couple of thousands of years, a new development emerged. Again, we explain it via a historical fiction.

At about 5400 BC there lived in a neighboring village a well-rounded thinker of that era. His name was Manu. He came up with a n ew idea for an intrinsic improvement. His acute insight led him to an improvement that partially dispenses with the intermediary tallying!

This seems to be a counterrevolution. But progress often is a result of conflicting factors. Manu always would say: “Why should we carry those pieces of wood or bones? They are too heavy to carry around and are liable to be lost too. “Look at our bodies! What do you see? You see fingers, hands, legs, toes, eyes and ears. Each of these can be utilized instead of those silly scratched sticks.”

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Moreover. He would always remind the doubters by saying:

“You always carry your fingers, hands, eyes, ears and feet with you wherever you go, and wha tever you do: Hunting, fishing, sowing seeds gathering “fruits.” In about 3700 BC the whole population of the country forgot all about the old tally sticks. They found themselves using the left forefinger to stand for ANY collection made up of a single element, two fingers to represent ANY collection made up of pairs, a closed hand to represent ANY collection of five elements, etc, These were given names and symbols on clay. Counting by using parts of the body generated few numbers and their symbols. We, in this book, will call those few numbers by poetic terms: “Baby numbers” 8 and “baby numerals.” Counting with the aid of parts of bodies has to be done in a systematic orderly manner; otherwise, confusion would result. Consequently, those baby numbers can be arranged in a natural order. Thus, ancient people began to realize that for every number there is a larger number. Hence, there is no largest number. Therefore, numbers cannot 8

Names that are more precise would probably be “atomic” numbers, and “atomic” numerals.

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end. That is, in the precise today’s mathematical terminology; the set of numbers constitute an infinite set. This conclusion might have been reached via another route. The ancient human being who possessed a herd of “many” sheep (that is greater than his baby numbers) used to notice that on several occasions his herd had increased by new births. Still they were “many.” Alternatively, he might see “many” trees so many that cannot be handled by the baby numbers he has. He used to notice that new trees had popped up. Still there were “many” trees. There were similar situations from which he could conclude that “Many” may be larger or smaller than “many”!9 Therefore, questions like the following must had arisen in the minds of some Mesopotamians and other people of ancient civilizations: How many is “many”? Is there a way to measure a “many” collection?

This sounds contradictory. In fact, the mathematical definition of an infinite set: A set is infinite if it can be made in one-toone correspondence with a proper subset of itself. Moreover, mathematicians know that there are hierarchies of infinite sets.

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Can the baby numbers be used to represent the collection of “many” objects?

Thus, ancient man was aware that his baby numbers are inadequate for counting every collection of things even if using further parts of his body to augment those numbers. Not all ancient civilizations (even the not so ancient civilization) were able to solve the representational problem in an adequate manner. The Greeks and the Romans did not. Let us pretend, for a moment, that we are ignorant with the current system of placevalue numerations. Then we would, most likely, be using the Roman numerals. We would have to write 4999 as MMMMCMXCIX. How many lines would be required to write down the budget of the United States? Can we imagine doing arithmetical operations with the Roman numeral? Though, the place-value system seems today so natural and so obvious, yet the great Greek thinkers, philosophers as well as mathematicians did not bother to examine the system, in spite of the fact they were exposed to it in Mesopotamia or in Egypt. Does not strike us as very strange? Is the reader aware of the fact that the transmission of the place-value system to Medieval Europe via the Arabs was met with great official resistance because of being from an “infidel” source?!

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Thanks to the Ancient East that made those Roman numerals to be presently exiled and almost exclusively used in modern society following the names of kings, queens, ships, and few covering pages of a book! The Mesopotamians, the Ancient Egyptians as well as other Eastern civilizations and Mayan civilization solved the problem by what is called today the placesystem of numeration. Their solutions were likely obtained independent of each other, but slightly different. The Egyptians used the decimal system, (which we currently use in our daily life), while the Mesopotamians used the truncated sexagesimal system. 04.5. THE PLACE-VALUE SYSTEM How did the ancient people invent the placevalue system? Again, we make a guess in the form of a history fiction:

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Nimrod was an ambitious young man, living in a village in the northern part of Mesopotamia. He, as almost everybody else, had learned to count with his fingers up to s ixty and had learned their standardized cuneiform symbols. Sixty was the end of the counting world! Every collection having more elements than sixty looked as if, it was beyond the ability of being counted and was banished by simply describing it as “many.” Although Nimrod was not happy with “many,” he did not give it much thought at the beginning. His instincts were more oriented towards trade. His aspiration was to get rich by trading with neighboring villages. Sometimes, there comes a moment in the tide of time or a critical situation in an individual’s life that diverts the trajectory of his or her life. Such a moment had come to our hero Nimrod. One day, Nimrod had plucked from his orchards in the fertile land of the North of Mesopotamia a very large collection of apples and other fruits too. He and his assistant were about to load the apples and the fruits in sacks on the backs of a herd of donkeys in preparation to travel to neighboring villages. But something stopped him. The large collection of apples seems to rise up and speak demanding to be counted. Nimrod imagined the collection of apples were screaming at him, “Don’t chicken out by the ‘many’ “. He thought these “silent voices” were the gods’ voices. He turned to his assistants “How many apples do you think we do have? “I d on’t really know” answered the assistant “It looks they are fairly many”.

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Nimrod angrily retorted back, “I do not want to hear that word again. The gods are ordering me to count. Therefore, let us obey the call of gods. If we do, we’ll gain some insight ” Thus, the two began to count the apples. “ONE, TWO, THREE, …, SIXTY” They finished their count “THAT IS GREAT”! Both exclaimed at the same moment. But there remain many more apples not counted yet. “Master,” said the assistance “May I ask your permission to count the rest of the apples. Nimrod nodded his head in approval. They put the already counted collection aside in the sacks and replaced it by a large mango. The two repeated the counting: “ONE, TWO, THREE, …, SIXTY!” And they kept repeating the count several times replacing every collection of sixty of counted apples by a mango. At last there remained only seven apples. Now, glancing at the collection of mangos the two observed that there were “many” mangos too. They did not utter the word. Nimrod turning to his assistance, asked, “You look tired. Aren’t you?” “No, I am not. I am only a little hungry” was the reply. “Then take an apple from the sacks over there on the back of that donkey; and let us finish the job by counting the mangos too” The two repeated the same process but only replacing every heap of sixty mangos by a melon. The counting process took them about an hour. At last they reached the bottom line. The count was: Two melons, five mangos and seven apples. Each mango represented sixty apples and each melon represented sixty mangos. 71

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Nimrod scratched on a wet clay tablet the three numbers in juxtaposition, [2] [13] [7] with no further comments. The assistant requested his master to describe the complete detailed story. But Nimrod intoxicated with his success in domesticating the “ many” answered in a categorical manner “NO”. The method very soon spread rapidly all over the country, like fire. It seemed that the ideas were “floating in the air” waiting only for some person to pull them down. Was the “floating in the air” synchronized with the voices of Nimrod’s gods? May be they were.

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Nimrod made it a habit to record the counts of the apples he gathered in each season. On one season he found the count was slightly odd, the count was thirty-five apples, three melons and NO mango. The emptiness of the mango pile, confused Nimrod and is assistant. The latter was tempted to record down [3][35]. “WAIT A MINUTE!” ordered Nimrod “THIS WOULD STAND FOR SOMETHING ENTIRELY DIFFERENT. ” Nimrod and his assistance, after some discussion, finally settled down by recording: [3] [35] Leaving a blank space for the non existent collection of mango pile. A great discovery slipped through Nimrod’s fingers. He could have invented or adapted an explicit symbol for the empty pile. However, the voices of Nimrod’s gods would not permit him to put anything in that vacant space. It was a “heresy” to put a thing as a representative of nothing whispered the voices. Though this is a history-fiction but something similar actually must had taken place. One even can make a joke that actual real history is a caricature of history fiction!

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We have already illustrated in the previous section the notion of the well-known decimal place-value system. This is not the only place-value system. In fact, the reader can “cook” his or her favorite place-value system. The cooking can be made by choosing a positive integer greater than one, called the base of the system and by choosing symbols for the finite nonnegative integers that are less than the chosen base. We call this set of symbols (as mentioned earlier) “poetically” as the “baby numerals” of the system. Then ANY nonnegative integer is expressible by juxtaposition of a sequence of the baby numerals. Each symbol would stand for different number according to its position. Negative integers can also be expressed by simply putting the minus sign “–“ on the left of the numeral representing the absolute value of the number (as everybody knows). From the logical point of view, every place-value system is as good as any other. However, some systems are more convenient than others for certain practical or theoretical purposes. Nimrod’s story can be laid on rigorous basis. Lest, the attention of the reader be diverted into rigorous arguments, we will present in APPENDIX (MATHEMATICS) the mathematical foundational theorem (without proof) as well as two examples of place-value systems having bases two and twelve.

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Place-value system is a great invention of the human race. Several other civilizations: Egyptian, Indian, and Chinese, most likely independent of each other made the same invention. The Mayan people invented the same system. Laplace (1749 â&#x20AC;&#x201C;1827) the great French mathematician remarked about the placevalue system: The method of expressing every possible number using a few symbols is ingenious. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated, its simplicity lies in the way it facilitated calculations and placed arithmetic foremost among useful inventions. The importance of this invention is more readily appreciated when one considers it was beyond the intellectual giant of antiquity Archimedes. Laplace made his remark referring to the ancient Egyptians invention of the placevalue numeration system; the Mesopotamian contributions were not known and not yet discovered at his time.

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04.6. TRUNCATED SYSTEM

SEXAGESIMAL

The following table gives “the babynumerals” in the truncated sexagesimal system:

Table 04.1. Table of the “baby” numbers of the Mesopotamian number system

The reader should notices that there are only fifty-nine symbols, not sixty. There is no symbol for zero. The first ten symbols

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consist of one symbol only. The other fortynine symbols are combinations of the first ten. For example looking at the numeral corresponding to thirty-eight one finds that it is made up of the symbol for ten repeated three times followed by a symbol corresponding to eight. This is the reason we describe the system as hybrid. Two shortcomings in the table must strike the reader: • The absence of a symbol for zero. • The hybrid method of generating the “baby-numerals” of the system. We will discuss these shortcomings in section 04.7. Meanwhile let use an illustrative example:

Figure 04.2

An example of a number in sexagesimal system

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The number depicted in the above figure, can be written in the sexagesimal notations as: 2 1 x (60) 3 + 57 x (60) + 46 x 60 + 40 This amounts in our decimal system to 424000. Since it is difficult to reproduce the original Mesopotamian symbols, we, in this book, will use the usual numerals with square brackets. That is; [0], [1], [2], [3], â&#x20AC;Ś [59]. Thus, the number in the above tablet can be expressed as [1][57][46][40].

04.7. PECULIARITIES MESOPOTAMIAN SYSTEM

THE

Let us clarify things from the beginning. The defects are definitely not in the sixty-base system itself, as some readers may think. In fact, as was mentioned earlier, the sexagesimal system is, logically as good and mathematically as sound as any other placevalue system. It is true that for modern people, who are brought up with the use of the decimal system, the sixty-base system definitely seems cumbersome and difficult to manipulate. However, enough practice with it

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would probably make its manipulation easy and painless even to modern people. The real defects of the Mesopotamian version of the sexagesimal system are the inadequacy of the symbols used. Those Mesopotamians seem that they had relied too much on the “understanding from context,” and overlooked the vagueness that may result. We will explain later that they could have avoided ambiguities if they had employed only three more symbols. We, in this book, will call the Mesopotamian version of the numeration system “truncated sexagesimal system.” We should point out that ambiguities, which will be discussed a little later, punch us on our faces because there is a time-bridge of four or five thousands years separating us from the Mesopotamians. We must understand that the scriptwriters were writing for their contemporaries who would understand many things from their background. Moreover, the scriptwriters might never have crossed their minds that their tablets would be read so many thousand years later. It is evident from the Mesopotamian contributions in many fields of knowledge, that they were skilled people. Had their contemporaries (who built the high ziggurats, dug the irrigation canals, predicted eclipses and wrote immortal literature) felt the existence of deep insurmountable ambiguities,

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they would have most probably made the necessary modifications. Incidentally, our modern society is not immune from similar practices. Now let us turn our attentions to the shortcomings as we see them today. The absence of an explicit symbol for an empty space-holder is an essential deficiency. This lack makes the written numerals ambiguous. For example 1, 60 and 3600 were expressed by the same symbol for one. Furthermore, the numbers 21, 201, 2001 would have been written 21, 2 1 and 2 1 respectively by leaving some vacant spaces between the digits. The use of vacant spaces is some sort of an implicit symbol for zero as a placeholder. Nevertheless, the widths of empty spaces are difficult to recognize or estimate especially on clay tablets. On the other hand, the lack of an explicit symbol for zero is less serious than was thought of at first encounter. Since the chance of having, a zero in the middle of a numeral in the sexagesimal system is about one sixth of its occurrence in the decimal system. Anyway, we pointed out earlier that the Mesopotamians, at the climax of their civilization, around 700 BC, seemed to have felt the need for an explicit symbol for zero. They did this in a rather â&#x20AC;&#x153;shy mannerâ&#x20AC;?. We will

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consider this shy modification in the next chapter. The fifty-nine symbols for the “baby numerals” in the Mesopotamian version are not entirely independent, but are, so to speak, decimally generated. Forty-nine of them are made up of two symbols. We describe this by saying that the “baby numerals” is a hybrid system. The hybrid property, in its self, is not a deficiency. In fact, in our modern society, we do use hybrid notation too. For example, we write 4 / 15 / 2005 having three hybrid independent systems of month, day and year separated by the separator “/”. We do not write 4152005. Some military people write 1345 for the time of the day. It is understood from context that it meant 1:45 PM. The shortcoming is resulted by the lack of clear indicators separating the different elements. This shortcoming in the Mesopotamian system may lead to further ambiguities. For example, the numbers 12 and 602 are expressed by the same Mesopotamian symbols. Another deficiency, in the Mesopotamian version, is the lack of a sexagesimal point, again leading to further ambiguities.

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In the decimal system used today,

there is a point “ ” in addition to the ten symbols used to represent “baby decimal numerals.” It is used to designate, as every one knows, a decimal fraction. For example:

the point in “7 5” makes “ 5 ” stand for “

5 ”. 10

And the point in “ 75” makes “7” and “5” to

7 5 stand for “ ” and “ ” respectively. It is ! 10 100 unfortunate that there corresponds no such point-symbol in the Mesopotamian version of the sexagesimal system. The following table ! ! provides examples of possible ambiguities that may arise because of lack of sexagesimal point. Number in the decimal notation

The number expressed as ordinary fraction in base b = 60

The number written in the sexagesimal system

What the Mesopotamian s would write

The number may be confused with:

.75

45 / b

[0].[45]

[45]

30 / b

[0].[30]

[30]

[0].[2][24]

[24]

144

.04

2 / b + 24 / b

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Table 04.2.

04. 8. IN SPITE OF THE SHORTCOMINGS â&#x20AC;Ś In spite of all those deficiencies, we emphasize by repeating that we should remember that the Mesopotamians were recording for their own contemporaries who comprehended the background and understood what was meant and what was required from context. Our modern society did the same in the near past. Many young readers are likely to be unfamiliar with the slide ruler. For a long time, less than half a century ago, the slide ruler was an indispensable instrument for the engineers and laboratory researchers. It was made up of two logarithmically calibrated rulers capable of sliding along each other. Sliding of the rulers added or subtracted the lengths. The sliding operation boils down to addition or subtraction of logarithms. Hence, they correspond to multiplication or division. With the advent of the calculator and the computer, it had become obsolete. Figure 04.3 shows a picture of the outdated slide ruler.

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Figure 04.3

Slide Rule

The slide ruler was used to carry out multiplications, divisions, and manipulations in trigonometry. For example multiplying 363 by 25 or 363 by 250 or 3.63 by 0.25 or 36.3 by 2500, would be carried out in exactly the same process. The accuracies of the readings are usually up to three or four decimal places. The reading of the above multiplications would be 986. The user has to mentally calculate the exact number of zeros to be placed to the right of the numeral â&#x20AC;&#x153;986â&#x20AC;? or the exact position where the decimal point is placed. Is this different from the Mesopotamian lack of an explicit symbol for zero or the lack of sexagesimal point? Not Much.

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In spite of all those deficiencies and shortcomings, which we discussed, the ancient Mesopotamian seemed to have used his head to obtain the required result just as our “pre-computer” engineer when using his slide ruler.

04.9. WHY WAS BASE SIXTY USED? The question “Why did the Mesopotamians use sixty as a base for their numeration?“ must already have arisen in the minds of many readers. However, before discussing the question, we refresh our memories of our inheritance from the Mesopotamians. Part of sexagesimal system persists even today in modern society. The year is divided into twelve months, the day into twenty-four hours, the hour into sixty minutes, and the minute into sixty seconds. The circle is, almost universally everywhere, divided into 360 degrees 10. A degree is divided into 60 minutes. A foot is made up of 12 inches. The zodiac is divided into twelve sectors. 10

However, the Russians divide a circle into 400 degrees. Thus a right angle is a 100 Russian degrees.

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Eggs are sold in dozens; beer is sold in six packs! There are some traces in our languages of Mesopotamian words (even in Western languages). For example the words “dozen,” ‘”Sabbath” These are some of the evident inheritance from the Mesopotamian civilizations. There are other less evident inheritances from them11 Back to the main question! There is no agreement among historians about the reasons that made the Mesopotamians use sixty as a base. In fact, there are several theories in dealing with the question. A widely spread theory claims that the reason lies in the fact that sixty has many factors, namely: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, making it convenient for division. Thus, it is a fact, that sixty-base is more convenient in simplifying representing inverses of some numbers, for example: 1/2, 11

A very interesting rarely known inheritance is the following: Many Arab singers begin their songs by repeatedly singing the phrase “Ya lail, Ya ain”. In modern Arabic language the phrase translates into “Oh night. Oh eye”. The songs have nothing to do with neither nights nor eyes. The singers without being aware are in fact repeating a six thousand years old religious Sumerian mantra hymn calling the Sumerian gods Aleel and Anu! What a magnificent inheritance!

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1/3, 1/9 and 1/32. These would be written as:

. [0].[2][13][20] [0] [30],

[0] [20],

[0] [6][40]

and

respectively. For more examples, see Table 06.2 in CHAPTER 06. Although it is, true that Mesopotamians carried out long division using their tables of inverses, yet the factor-theory for using sixty as a base does not seem convincing. It presupposes the existence in Mesopotamia some sort of a scientific council that weighs the merits of different systems and decides accordingly. There is no indication that there was such a council. Even if there were such a council or a committee, base twelve seems to be more appropriate. A place-value system of numeration with twelve as a base has the same advantages as described above. In fact, there are today societies that call for adopting such a system. For discussion of the duodecimal system sees APPENDIX (MATHEMATICS).

The famous historian of ancient mathematics Otto Neugebauer had a different theory. He claims that there was an older place-value numeration system with base ten. The old system was modified in order to divide measurements (of length, area, volume, time, weight) in an easy way in three, six, twelve, thirty parts. This theory might be true. Again, it requires the existence of a body of people

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whose responsibility was the decision to change from one system to another. Moreover, the division of measurements itself was more likely the result of the system of numeration, not the other way around One should remember the resistance that took place a few years ago in the United States to the change to metric system in spite of the existence of a committee that recommended the change. E. M. Bruins, the Dutch mathematicians and assyrianist, suggested (in a private conversation) that a unit of a fighting force was made up of thirty men; during action in a battle they are given twice the ration of food. Therefore, sixty is a base. The question is asked why a military unit was made up of thirty. E. F. Robertson in [OR 2] made the following suggestion, which seems plausible. Let us follow the chain of his reasoning. The way people count determines the base of their numeration system.

This seems to be an acceptable assumption. People - as we have seen - count with the aid of parts of their bodies.

There are civilizations in which people, count with the fingers of their two hands. Those people tend to use the decimal system. There are people of other civilizations who count with the fingers of one hand. Those

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people tend to use the five place-value system. Others count with the fingers of their hands and toes12. Those people tend to adopt the vigesimal system that is the place-value system based on twenty. Robertson conjectured 13 that the Mesopotamians (wishing to count as far as possible with the parts of their bodies) counted by using the fingers of both hands in the following peculiar way. They count by successively bending down the three parts of each of the four fingers of one hand except the thumb. When all the fingers of that hand are closed, they had counted up to twelve. Then they close one finger of the other hand, open the first hand. Repeating this process until the second hand is completely closed, they could count up to sixty. This seems to the author to be a good explanation for the choice of sixty as a base. The word “dozen” often used in several modern languages is in fact a word “corrupted” from “durzan”, which is a word, borrowed from one of the Mesopotamian 12

See Chapter 05 for a very brief account on the independent contributions of the Mayan people of Central America. 13

Science, according to the philosopher, Karl Popper starts with conjecturing.

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languages. It means “one fifth.” That is one fifth of the base sixty. Moreover, it might have meant a “hand” too. We can imagine a Mesopotamian farmer counting, say, the apples in his farm. When his left hand is closed, he shouts “one dozen,” then continues the counting process “two dozens”, up to “five dozens”. Robertson’s conjecture is left for archeologists to refute it or support it.

04.10. WHY WERE NUMERALS” USED?

HYBRID

“BABY-

Again, no one knows for sure. But, it seems likely that at an early period of the Mesopotamian civilization when the writing was not yet crystallized, there were two factions of people, one the original Mesopotamians, the other might have been immigrants or invaders. One of the groups counted on the fingers of both hands; the other used the two hands method of counting as described in the previous section. As writing gradually emerged and developed, the two systems of recording numbers flowed together into a confluence of a hybrid written system.

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Again, this is a conjecture, which is left for archeologists and historians to corroborate or falsify. The French number-words reflect a trace of a confluence of two systems: the decimal and the vigesimal (i.e. twenty-based system) Here are some of those traces. Dix (ten), Vingt (twenty), Trente (thirty), Quarante (forty), Cinquante (fifty), Soixante-dix (sixty), Soixante-dix (sixty and ten, that is seventy)

Now, note the monotonic change in the following: Quatre-vingts (four twenties, that is, eighty) Quatre-vingt-dix (four eighties and ten, that is ninety.)

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05. Detour in Zero's History

Zero is an important number. Never underestimate its importance. â&#x20AC;&#x201C; An Iraqi High School mathematics teacher

05.1. INTRODUCTION This chapter, as its title indicates, deviates from the main stream of the book. It consists of an anthology of subjects, revolving mainly, but not exclusively, around zero. We will deal, among other topics, with the

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questions that may have already been arisen in the minds of readers. Namely: What had inhibited the Mesopotamians from having zero in their numeration system? Which civilizations that independently developed the concept of zero? Which path the concept of zero traveled in its slow acceptance as a number?

Zero was and is a tantalizing concept. It had been conceived in one place and was born in another; it had passed through many great minds and crossed most diverse geographical borders. One may think that the concept of zero, after its â&#x20AC;&#x153;shyâ&#x20AC;? appearance in late Mesopotamian civilization, would immediately enter into the stream of mathematics from that time on. History shows that it did not. We, in this chapter will also dispel some widely accepted wrong notions that prevailed and still prevail in modern society concerning division by zero. These topics will take us into diversions in the history of zero, inside and outside the Mesopotamian land and its civilizations.

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05.2. TWO CURRENT USAGES OF THE SYMBOL “0”

The symbol “0” has, in our present day modern society, two current usages: First. The symbol “0” is used to stand as an indicator for an empty-place in our place-value number system. In this capacity we say: 0 as a placeholder for an empty place. For example: Instead of writing “3 5” as the Mesopotamian used to do, we put 0 into the vacant space and write “305”. Instead of writing “2 5”, as the Mesopotamian used to do, leaving a wide vacant space, we put 00 and write “ 2005”. Instead of writing “2 5 ”, as the Mesopotamian used to do, leaving vacant spaces, we write “2050”.

Second. The symbol “0” is also used as a numeral representing a “respected” number that can be arithmetically manipulated. It can be added to other numbers or to itself. It can be subtracted from other numbers or from itself

For the moment, we are not dealing with the problem of dividing a number by zero. We leave it for section 05.10.

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It can be multiplied by other numbers or by itself. It can be divided by other numbers.

The distinction between the two usages is significant. Thousand of miles and nine centuries separated their “space-time” emergences. We feel that some readers are unhappy. We seem, as though hearing their reproaching murmuring, “How come, you modern mathematicians, so fond of rigor, become so illogical when you represent two different notions by the same symbol?”

We do concede that there is a certain lapse in logic. To err is human; to blame it on somebody else is even more human. We put the blame on our mathematical ancestors. Nevertheless, in spite of the lapse in logic (whether due to modern mathematicians, or to our mathematical ancestors) no harm ever resulted. In order to demonstrate that no consequential wrong ever happened, let us, for a moment, have different symbols for zero as a placeholder and for zero as a number.

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Let us denote the number zero by the symbol “Ω” and keep the symbol “ 0 ” as standing for a placeholder. It is obvious that any statement known to hold true for the number zero would also be true for Ω. In particular, Ω + any number z = z, and Ω x (any number) = Ω. Consider for example the pair of numerals 7Ω3 and 703. The numeral 7Ω3 stands for 7x100 + Ωx 10 + 3 = 7x100 + Ω +3 = 7x 100+3. Since 0 is a symbol for placeholder, the numeral 703 stands for 7x100+3. Therefore, 7Ω3 and 703 represent the same number. A similar procedure can be carried in general to show that using the same symbol to represent the two different usages for zero, though illogical, is harmless. After these clarifications, we turn our attention in the next section, to a crucial question: Why was an explicit symbol as a placeholder persistently absent for thousands of years from the Mesopotamian numeration system?

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05.3. WHY THE ABSENCE OF “0”? There is a wide spread opinion that the Mesopotamians did not have any idea about the concept of zero. We do not completely share this opinion. It is true that there are no indications that the Mesopotamians ever used zero as a number in its own right. It seems that they were not ready for that yet. On the other hand, we believe they had a good idea about zero as a placeholder. The mere leaving of vacant spaces between some of their “baby numerals” is an indication that they did know about the notion of a placeholder. The vacant space is some sort of representation, a foggy representation. Nevertheless, such representations are vague, difficult to notice, hard to measure or to estimate their widths; consequently led to confusions and ambiguities. Were the Mesopotamians aware of the possible haziness? We are not sure, but we bet they did. Therefore, the question can be rephrased: Why the Mesopotamians did not use an explicit unambiguous symbol for a placeholder?

A partial answer is the fact - mentioned in the previous chapter - that the chance of

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encountering a zero as a placeholder in the sexagesimal system is one sixth of the chance of encountering a zero in the decimal system. Thus, the problem is less acute than it seems to be at first sight. Though, this answer, has some bearing on the phenomenon of absence of an unambiguous symbol for a placeholder, it is an inadequate explanation. We believe that there was a much more profound reason that inhibited Mesopotamians so persistently for thousands of years, from using a clear representation for zero as a placeholder. What was it? It seems likely that they had a psychophilosophical mental complex blocking against adopting an explicit representation of zero. It is: A heap consisting of nothing, (encountered in a counting process), cannot be represented by a symbol, which is a thing. This needs an explanation. If we remember that the Mesopotamians began their writings as pictograms, then it is clear that abstract ideas had roots in tangible things. As an example, the collection of three fingers, (being among the collection of all trios), may be taken as a representative of them all. Thus, the abstract number three is

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represented, in drawing, by three fingers and later became three bars. Three fingers are material things. Their representation on clay tablets by III (or something similar) is also a material thing. On the other hand, an empty set has no element. A Mesopotamian must have some familiarity with the notion of what we call today an empty set. If his children ate all the apples that he kept at home, the set of apples left is empty. If on one morning he found all his poultry were dead (they were stricken by some sudden disease), the poultry left is an empty set. If on returning back home he found no children there, he began to worry; the class of his children at home was empty.

The roots of the Mesopotamian writings established frameworks of minds that inhibited them from representing a collection consisting of nothing by a thing. For them to do otherwise is not only, illogical, but also a taboo. In other words: They were prisoners of their own framework of mind. The only thing a Mesopotamian can do in handling a situation like our “305” is to represent it by a blank space such as “3 5”. That is what they did. We believe that this is the deep-rooted reason that inhibited the Mesopotamians for

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thousands of years, (until about 700 BC) from inventing an explicit symbol for a placeholder. We may even conjecture that Mesopotamians were well aware that the method of their writing numbers led to ambiguities. They might even have thought that the existence of ambiguities is the price that must be paid in order to be logical. Understanding from context would take care of any possible ambiguity. Or as the rephrased famous old cold-war slogan goes “Rather be ambiguous than fallacious.”14 It is worth repeating the last paragraph of Nimrod’s history-fiction narrated in the previous chapter, though it is a fiction it reflects the idea clearly:

The rephrasing of the slogan: “Rather be dead than red.”

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Nimrod made it a habit to record the counts of the apples he gathered in each season. On one season he found the count was slightly odd, the count was thirty-five apples, three melons and NO mango. The emptiness of the mango pile, confused Nimrod and is assistant. The latter was tempted to record down [3][35]. “WAIT A MINUTE!” ordered Nimrod “THIS WOULD STAND FOR SOMETHING ENTIRELY DIFFERENT. ” Nimrod and his assistance, after some discussion, finally settled down by recording: [3] [35] Leaving a blank space for the non existent collection of mango pile. A great discovery slipped through Nimrod’s fingers. He could have plugged something in the vacant space and he would have invented an explicit symbol for the empty pile. However, the voices of Nimrod’s gods would not permit him to put anything in the vacant space. It is a heresy to put a thing as a representative of nothing whispered the voices. It needs a great effort to get rid of a psychological complex.

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In case of individuals, electric shocks sometime treat complexes. What about cultures? 15

05.4. POST 700 BC Beginning about 625 BC a new youthful empire ruled the land of Mesopotamia, after a long civil war. This young empire began to expand and invaded Syria and Palestine. It might have happened that people in Syria or Palestine did not understand from context, which put some scriptwriters out of the usual framework of mind. Wars and territorial expansions have impacts on societies similar to the electric shocks on individuals, for example: â&#x20AC;˘

â&#x20AC;˘

Eventual defeat of Crusaders led to the religious reformation in Europe. The 9/11 terrorist attacks was a shock to the American

Gary Gardegree mentions in his paper [Ga] that the Egyptian numeration system does not require a symbol for zero, yet the Egyptians nevertheless had a symbol for zero, which they used for a variety of engineering and accounting purposes including projects, such as the building of Pyramids!

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administration that made drastic changes in its policies .

Back to the Mesopotamians! The Mesopotamian civilization reached its climax about 700 BC. The burden of the psycho-philosophical complex that was described in the last section started to be felt. They seemed to begin to realize it is better to relieve their tablet readers from the burden of understanding from context. The Mesopotamian of that period began to use the semi-quotation mark (“) as a symbol for a placeholder in a rather shy manner. They put something like that as a symbol for the empty space between some of their “baby numerals.” Therefore, the post-700 BC Mesopotamians invented an explicit symbol as a placeholder. There are excavated tablets dated to this period showing numerals, for example, 7”4 and 7””4 for 704 and 7004 respectively. Mesopotamians were not completely healed from the psycho-philosophical complex since they were still clinging to write 74 for both 740 and 7400, and still adhering to understanding from context. We described their half-hearted usage by as shy. This clinging may seem strange and silly to us in modern society. Nevertheless, it should not. Modern society still uses “understanding from context.” For example, if you ask for the price of a magazine in a

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bookstore and being told that it is two, you understand it is two dollars. However, when you ask for the price of an airway ticket to a nearby town and you were told it is two, you understand it is two hundreds dollars. If you go to a car dealer and ask for the price of a pre-owned car and you were told two is the price, you would immediately understand it is two thousand dollars. Anyway, the Mesopotamians had just partially overcome their mental barrier. It might have been possible for them to be completely free from it; had their empire survived some few centuries more than it actually did and they might even have reached the concept of zero as a number. Alas, the empire soon collapsed when the Persians under King Cyrus II invaded Mesopotamia in 539 BC and made it a Persian province. In order to make quick surveys (in space as well as in time) of the works of other people are and will be doing mainly, but not exclusively about, zero let us make a imaginative journey leaving Mesopotamians for a while; we will come back to then in the next chapter. Our first stop in our journey will be in Greece where there are indications of a new great civilization.

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05.5. ANCIENT GREEK ROLE: “GREEKNUMBER-GATE”

About 1000 BC, a few centuries before the collapse of the Mesopotamian Empire, and when the Mesopotamians had just begun using a symbol for a placeholder, a new civilization was beginning to flourish on the Greek shores of the Mediterranean. The majority of Greek people were not farmers, attached to their lands, as most Mesopotamians were. They were mainly merchants traveling around the world and building strong trading ties with the Egyptians and Mesopotamians. In fact - as was mentioned in a previous chapter - the name “Mesopotamia” was coined by Greek geographic-tourists. Moreover, the source of some of the Mesopotamian’s history came to us from Ancient Greek geographers. For example, our knowledge of the famous Babylonian hanging gardens did NOT reach us from excavated tablets but from a Greek tourist who carefully described those hanging gardens. Few centuries after the collapse of the Mesopotamian Empire, Alexander the Great (356 BC – 323 BC), (who conquered practically the whole known world at that time and defeated the Persians who occupied Mesopotamia) was very much fascinated by

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every aspect of Mesopotamian culture. In fact, his fascination was so great that he ordered or recommended to his military commanders that they get married to the beautiful Mesopotamian women! They did. Moreover, he himself, leaving his first wife in Macedonia, took for himself a second wife from Mesopotamia! Thus, the Greeks were well acquainted with the ancient Mesopotamian civilization before and after its epic and had extensive contacts with the Egyptians and the Mesopotamians. They learned a good amount from both. Some of the Greek visitors to the two ancient countries turned later to be famous well-renowned great thinkers. Thales (about 624 BC -about 547 BC) and Pythagoras (about 569 BC - about 475 BC) were among the famous visitors. Therefore, it can be concluded that the Greeks must have been familiar with the place-value systems used in both Egypt and Mesopotamia. It is well known that Greek civilization produced great minds in most fields of knowledge that contributed immortal results, having impacts, throughout all ages, on most disciplines of knowledge. Even today, many novel philosophies or new scientific theories have some traces that go back to the Greeks.

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In spite of all those cultural contacts and in spite of all those great thinkers including great mathematicians and philosophers, the Greek numeration system was primitive, and remained primitive for more than a thousand year. The Greeks used an alphabetic numeration. To the best of our knowledge, none of those great minds ever recommended an updating or overhauling their system of numeration. Those thinkers - through the many cultural contacts with the Egyptians and the Mesopotamians - should have been well aware of the superiority of the place-value system over the alphabetic system. Yet, they stayed aloof and did not bother themselves with numeration systems. This detachment of the Greek great thinkers from dealing with arithmetic that persisted for centuries all throughout the period of their great civilization indicates the existence of a deep cause. We noted in the Section 05.3 that the Mesopotamians had psycho-philosophical complex that blocked the development of putting a symbol for a placeholder. The Greeks also seemed to have a deep-seated complex that turned them away from arithmetic and algebra altogether and caused them to concentrate mainly on geometry and made them to look down upon arithmetic.

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This calls for an elaboration of a possible cause. Pythagoras was a son of a merchant. During his childhood, he accompanied his father in the latterâ&#x20AC;&#x2122;s commercial journeys to Ancient Egypt and Mesopotamia. Later as a young man, upon the advice of his teacher Thales, Pythagoras visited both Ancient Egypt and Mesopotamia just to study and learn from the sources of wisdom and knowledge of that time. The young Pythagoras learned many novelties from the two ancient civilizations. He learned their mathematics, music, astronomy, religion, philosophies, and personal spiritual life. It is possible that through the influence of Egyptian and Mesopotamian priests he became a number-mystic. Later Pythagoras, on returning to his homeland its neighboring countries like Italy, and Crete, founded a semi-religious and semiscientific society, which included both men and women, and acted with a code of secrecy. The society was devoted to propagating his philosophy and his teaching about the Universe and about rebuilding personal principles on new standards. As a number mystic, he and his school partitioned numbers16 in several sorts of

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classes, for example: masculine and feminine numbers, beautiful and ugly numbers, perfect and non-perfect numbers amiable pairs, â&#x20AC;Ś etc. He believed that the deepest level of reality is mathematical in nature. He and his school believed that the Universe is explainable by integers and their ratios only. That is, by rational numbers. These numbers can be expressed as p/q where both p and q are integers and q is different from zero. His theory was derived from the harmony of musical scale. The Pythagorean philosophy had an immense influence on all the thinking of Greeks throughout the period of their civilization. This does not mean that every Greek thinker adhered or adapted the Pythagorean Philosophy. It means that a framework of mind at least for mathematics, was created which determines what issues are important to be investigated and which are not, which problems are worth studying and which are not worth studying17.

The word number in this chapter stands for positive integers unless otherwise stated with a preceding adjective. 17

This is what Thomas Kuhn calls a paradigm.

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What all these have to do with the deep-rooted collective psychological complex of the Greeks? It is an irony of fate that the very famous theorem of geometry that today bears the name of Pythagoras (though known to the Mesopotamians and ancient Egyptians some thousand years earlier) destructed his philosophy of the Universe. Every high school student knows the statement of that theorem. Let us refresh our memories. The theorem says: The square of the length of the hypotenuse of a right angle triangle equals the sum of the squares of the length of the other sides. This is the theorem that brought down Pythagorean philosophy. How? Consider a right angle triangle, in which the lengths of the two sides of the right angle equal one. Therefore, according to the theorem the square of the length of the hypotenuse is 2. Hence, its length is the square root of 2.

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Hypotenous =

12 +

1 By virtue of the Pythagorean Philosophy, the hypotenuse has to be a rational number. The crucial thing is: it is easy to prove that there does not exist a rational number whose square is two. This important statement can be easily proved18. 18

The proof runs like this: Assume there exists a rational number p/q where p and q are integers such that (p/q) 2 = 2. That is p2 = 2 q2 â&#x20AC;Śâ&#x20AC;Ś(1) We may assume further, without any loss of generality, that there is a no positive factor common, to p and q other than 1. Since if there were, they could be cancelled out. Now, p is either odd or even. The number p cannot be odd because if it were then the left-hand side of equality (1) is an odd number,

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1 = 2

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Its proof is so elegant and so simple that every literate person can follow. The proof is provided in the footnote. The proof amounts to conceding, contrary to the Pythagorean philosophy, that there DOES exist things in the Universe that cannot be expressed by rational numbers. The existence of numbers that are not rational was so to speak the “Number-gate” with respect to the Pythagorean School! It shattered the very foundation of their philosophy. The Universe after all, cannot be completely explained by rational numbers. The collapse of the philosophy was so devastating that the society tried to keep the existence of numbers that are not rational from the public. There are even some legends of murdering one of their members that revealed the secret.

while its right-hand side is an even number. This is obviously not possible. Therefore, p must be even. Thus p is of the form p = 2m … (2) Plugging equality (2) in (1) and simplifying one obtains: q2 = 2 m2 … (3) Repeating the same argument to q in equation (3) leads to q must also be an even number. Hence, contradicting the fact that there is no common factor between p and q. therefore, there does not exists a rational number whose square is 2.

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The members of the Pythagorean School would not hear nor speak about numbers that are not rational. This is the reason that these numbers were described as “surds.” The term is still in use when describing those numbers! This breakdown of that philosophy made a great change in the attitude of the Greeks toward mathematics. The Greeks instead of abandoning the philosophical framework and try to suit that framework to the actual universe, they abandoned dealing with numbers and arithmetic. All the trouble – from the perspective of the framework of mind of the Greek thinkers originated from numbers. Hence, they would think: ”Let us not bother ourselves with numbers any more. Numbers are agitators! It is better to avoid dealing with troublemakers and their representations!” Almost all Greek mathematicians 19 and philosophers turned away from the use of numbers and algebra. Not only were they not interested in numbers but also they looked down upon the usage of numbers. Numbers were not worth the thought of great minds. 19

Diophantus (About 624 AD - about 284 AD) was an exception. He was a Greek Mathematician living in Egypt Alexandria. Sometimes, he is described as “The father of Algebra.”

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Those great minds did not use numbers unless they had to. Nevertheless, the great Greek minds have to use numbers for down-to-earth purposes, for example, in their business dealings and in recording purposes that were needed for in commerce and other documentations. For those purposes, the ordinary Greeks as well as the great Greek thinkers used their alphabets as numerals. These numerals were not worth a minute of their minds! Just like astrology, alchemy and magic are not worth a minute of present day eminent scientists. Thus, the breakdown of the Pythagorean philosophy turned the Greek mathematicians from arithmetic to geometry. In geometry, numbers (even the surds that the Pythagoreans brotherhood was not allowed to hear of or talk about) are nothing but lengths of line segments; there is no need to name the numbers or to express them by symbols. By the use of a straight edge and a compass, one can perform all arithmetical operations of addition, subtraction multiplication, division, extracting square roots, solving linear and quadratic equations. The Greek mathematicians were marvelously successful in their pursuit. They made geometry a magnificent logical edifice. The study of geometry gave rise to several unsolved problems, for example, the problem

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of “squaring the circle” and “trisection an angle”, and the independence of the axiomatic system, which occupied mathematicians for about two thousand years and were not settled until the nineteenth century and eventually gave rise to Galois theory and nonEuclidean geometries. But, they neglected algebra, arithmetic and the adoption of an efficient numeration system. In conclusion, the failure of Pythagorean philosophy of the universe might explain - in our view - the reasons for the Greeks attitude of not adopting the placevalue systems. However, there were few exceptions. Mathematicians who were involved in recording astronomical data had to use numbers. In fact, Ptolemy in Almagest, working in Alexandria in Egypt, wrote around 130 AD using the very Mesopotamian sexagesimal symbols with an explicit symbol for a placeholder between the digits as well as at the beginnings of numerals. Nevertheless, zero, as a number in its own right, did not appear yet. We have to wait for daring minds that were free from collective psycho-philosophical complexes.

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Let us proceed with our space-time journey to other places and other times and visit the Indian subcontinent.

05.6. THE ROLE OF THE ANCIENT INDIANS Our journey will take us back in time to Asia. Our stop will be very brief. Along the great Indus River there flourished a great civilization, few centuries after the beginning of the Mesopotamian and Egyptian civilizations. The Ancient Indians contributed to science, religion, astronomy, and medicine. They predicted the occurrences of solar eclipses. Their science was always linked with spirituality. As far as numeration is concerned, they developed a decimal place-value system. Manuscripts dated about 200 AD, showed that they used as an explicit symbol a placeholder. Some historians claim that this was an independent Indian achievement; others state that it was borrowed from the Mesopotamian civilization at its peak. The latter is a possibility

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since Alexander the Great invaded both Mesopotamia and India. Armies are good carriers of ideas and diseases! There is evidence that a dot had been used in Ancient Indian manuscripts to denote a placeholder 20. India being a large subcontinent, uniformity in symbolism was lacking. Different localities used different symbols. At about 860 AD, some Indians used a symbol similar to the familiar “0” to represent the empty placeholder. Anyway, whatever the Indians used at that time, it was a symbol for the empty placeholder. It did not represent a “respected” number to be manipulated in the four standard arithmetical and other operations. At that time, nobody cared to adjoin an empty set to a set of three trees or remove an empty set from a class of five soldiers, or to distribute empty collection of apples to one’s children. Thus, empty set did not seem to possess the intrinsic natural characteristic in order to be “elevated” as a candidate to be a number. BUT! After an elapse of several centuries and after an extensive usage of numbers, a gradual new trend appeared. The new tendency was to dissociate further the notion 20

The dot is used today in the Arabic World and Iran

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of numbers from being attributes of collection of things to be THINGS themselves! In general, if a person “plays” long enough with an abstract concept, it becomes more domesticated, more tangible to that person. When that stage is reached, the person would dare to ask questions concerning that abstract concept he never imagined to ask before. Watch an enthusiastic professor of mathematics, lecturing on a very abstract mathematical concept. Watch the movements of his hands and smiles on his face. Hear his voice rising and falling. You will feel as though he is touching and seeing that abstract concept, and even kissing it and falling in love with it! When such a stage is reached, the professor would deal with that concept in a different way. He might investigate a situation (mathematical or not) that had seemed absurd or illogical before. Moreover, when he does, it often occurs that such an investigation is quite fruitful. Therefore, this was true with some Indian mathematicians and numbers. Numbers began to “live” an independent life of their own in the collective minds of mathematicians. Some daring minds began to ask what the properties of these

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abstract entities are. They found the following are some of their properties: There are “even Number,” There are “odd numbers,” There are “triangular numbers,” There are “square numbers,” There is a method of manipulation of numerals, which provides the sum of the corresponding numbers. There is a method of manipulation of numerals, which provides the product of the corresponding numbers. Numbers can be factorized, There are “prime numbers”, Any nonempty collection of numbers has a least element. … Etc.

Though numbers are abstract concepts, but to those daring minds, numbers would gradually seem almost as tangible as the as the houses over here, as the trees over there; they, like trees are bearing mentally tasty fruits. Some of fruits seem to be forbidden fruits. Nevertheless, many of those forbidden fruits will prove productive and have applications in different fields of knowledge. This “domestication” of the abstract numbers induced Indian mathematicians to pose questions to themselves and to others,

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which they never imagined to pose before. For example: What happens if an empty set is adjoined to a collection of things? What happens if an empty set is adjoined to an empty set? What happens if an empty set is detached from a collection of things? A group of soldiers went to a battle. Every soldier in the group was killed. This is an instance of: what happens if a collection is detached from itself? There is a group of warriors, and there is no food to distribute to them. What would be the share of each? … Etc.

About the middle of the seventh century AD some mentally adventurous Indian mathematicians, like Brahmagupta (598 AD 670 AD) and others attempted to answer these and similar questions. A new “embryonic” concept was beginning to be conceived in the minds of some Indian mathematicians. This embryo is nothing but the concept of the number zero which can be arithmetically manipulated just like other numbers. No one knows when that concept was exactly born, but it gradually was being admitted to be a respected citizen in the “republic” or “kingdom” of numbers.

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Brahmagupta and his contemporaries started to manipulate this new number. Their answers were magnificently correct (almost). They found that for every integer n: n + 0 = n = 0 + n; n x 0 = 0 = 0 x n; n â&#x20AC;&#x201C; 0 = n; 0/n=0 However, they had gone astray when they considered the problem of division by zero. This is not bad for the first people that we know of who tried to extend arithmetical manipulations to zero and negative numbers. Some five centuries later, other Indian mathematicians tried to update Brahambugtaâ&#x20AC;&#x2122;s result about division by zero. However, they messed up too. This is not strange, knowing that so many modern scientists still mess up. Why such a persistent failure in the operation of division by zero? The answer will be given in section 05.10.

Meanwhile we continue our journey across the vast space-time.

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05.7. THE CIVILIZATION ROLE

ARABIC-ISLAMIC

While Europe was living in its dark ages, the empire ruled by the Arabs from the eighth to the thirteenth centuries AD, stretched from Central Asia to Spain. The Arab-Moslem civilization reached its zenith during this period. The scholars of the Arab-Moslem civilization knew about science and arts more than any other of their contemporaries. The Arab rulers encouraged and facilitated translations into Arabic, of many classical works of literature, philosophy, and science mainly from Geek, Indian languages, and Aramaic (the language spoken by the late Mesopotamians and other parts in the Middle East). By the way, the Mesopotamians in spite of the collapse of their ancient empire remained for several centuries, sources of culture that illuminated Arab-Moslem scholars. The Mesopotamians preserved those great books and helped in their translations Into Arabic. There were great many scholars in the Arabic-Islamic civilization that made first-rate contributions in the advancements of science, geometry, algebra, philosophy, and literature. Those contributions had a great impact on Western science. However, we will not

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elaborate on those contributions since this section is devoted to the participations of some Arab-Muslim scholars in arithmetic, particularly in the newly born concept of zero. Foremost among early Arab-Muslim scientists who contributed to the advancement of arithmetic are Al-Khawarizmi (about 780 – about 850) and Al-Biruni (973 - 1048). Among the vast contributions of AlKhawrizmi in science, mathematics, and philosophy, is his book on Hindu-Arabic numerals, whose English title is “AlKhawarizmi on the Hindi Art of Reckoning.” The title of the book gave rise to the word, used extensively in modern times namely, “algorithm,” which is a corruption of the name of the author of that book. The work describes the decimal place-value system, which was a recent arrival from India and elaborates on it in a systematic manner. It explained for the first time the number zero (in Arabic, “Sifer” which means empty.) Al-Khawarizmi presented in his work methods of arithmetical calculations, and introduced a simplification of the processes of multiplications, divisions and methods of finding the square and cube roots. During the early parts of the eleventh century, Al-Biruni made several visits to India where he made a detailed study of Indian mathematics, astronomy, geography and religions. He wrote several works on Indian mathematics and science. He was familiar

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with the different ways the Indians symbolized their â&#x20AC;&#x153;baby numbersâ&#x20AC;? from different regions of the vast Indian subcontinent. The two main regional symbolisms were very similar to our current symbolism used all over the world. Both of these systems of symbols are used today all over the 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, and

world, the latter version is used mainly in the Arab world and Iran.

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We proceed with our space-time journey to Medieval Europe.

05.8. TRANSMISSION MEDIEVAL EUROPE

ZERO

During the medieval times, the Greek and the Roman numerals were in use in Europe. It is unknown when exactly the new place-value number system first came into Europe. It must have come gradually through several channels and appeared in some circles only to disappear soon. The oldest dated European manuscript containing ArabicHindu numerals was written around 1000 AD, when Pope Sylvester used them in some of his writings. Incidentally, Pope Sylvester was a mathematician, too. Leonard Fibonacci21 (1170 - 1250) is historically credited more than any other 21

Best remembered today by the sequence bearing his name. It is a sequence whose first two terms are 1, 1 and each term after the second is the sum of the preceding

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person for bringing to Europe the knowledge of zero, decimal system, and algebra. He was a great amateur mathematician who is famous for the sequence bearing his name. Fibonacci was working as a merchant in the Middle East, North Africa and Turkey (which was called at that time Anatolia) on behalf of commercial interests in Pisa and Florence. He was well versed in the Arabic language. The books of Al-Khawarizmi came into his hands. He immediately appreciated their importance and began translating them into Latin. In 1202 AD, he wrote a book “Liber Abaci” which means “The abacus book” in which he described the nine Arabic-Indian symbols 1, 2, 3, … and 9 together with the sign 022. Fibonacci, being a merchant, his work quickly gained the notice by Italian and German merchants and bankers, especially the use of zero. Accountants knew that their ledgers were balanced when the sum of positive and negative amounts of their commercial two. Thus we get: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … Etc. The sequence appears in different fields of mathematics and sciences. 22

It is worth noting that even Fibonacci was not bold enough to treat zero in the same level as the other nine numbers, since he described zero as a “sign” while he described the other nine symbols as numbers.

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transactions is zero. However, governments still were reluctant and suspicious of the Arabic numerals because of the ease in which it was possible to change one symbol into another. The Church was in doctrinal opposition to Islam and contributed to the denial of the use of “the infidel numbers”! Though outlawed, the merchants continued to use “0” in encrypted messages. Thus, the derivation of the words “cipher” and “decipher” meaning code and decode respectively etymologically coming from the Arabic word “Sifer” for zero. One might have thought that the progress of the number systems in general and zero in particular would have been steady from that time on. However, this was far from the case. Even three centuries later, the Italian mathematician, Cardan (1501-1576) was solving equations of the third and fourth degrees without using zero. He would have found his research much easy if zero was used. It was not part of his mathematical framework of mind. By the seventeenth century, zero as a number, was extensively used but still was running into some resistance.

Our journey will soon end at modern times, but before reaching that terminal let us have only a

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very quick peeping glimpse into the Mayan civilization flourishing on a yet unknown continent.

05.9. THE MAYAN CIVILIZATION The Maya people lived in Central America, in a region, which today is called Guatemala and southern Mexico. Their civilization is old but flourished in the period 250 AD and 900 AD. Among the civilization that developed the place-value number systems, the Mayanâ&#x20AC;&#x2122;s stands out, because its contributions are definitely independent of the others. The remarkable thing about this civilization is that the Mayan people had used a symbol for zero long before their development of the place-value system. Alas, they did not have a chance to influence other people. The place-value system they developed is vigesimal. That is, a system whose base is twenty. Thus, we conjecture that they might have counted on their fingers and toes!

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05.10. THOU SHALT NOT DIVIDE BY ZERO! Here is an interesting joke, which should always be remembered whenever a person is tempted to divide by zero: A mathematician’s house was on fire. The investigating fireperson asked how did it happen. The mathematician answered, “I accidentally divided by zero. Suddenly the house burst into flames.”

Many of today’s scientists cannot bring themselves to admit the fact it is impossible to divide by zero. You can divide by any number except zero. The proof of the impossibility to divide by zero is so simple that any high school student (or even a grade school graduate) can easily follow. It is advisable that everyone should be familiar with it. Therefore, it is tempting to include the proof in this section. For any number z (whether integer or not), it is known that (1) z x 0 = 0 = 0 x z, (2) zx 1= z =1x z. Let us for a moment and for argument sake, assume that there is a multiplicative

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inverse for zero, which let us denote by W. This means that W x 0 = 1 and 0 x W = 1. Therefore, we have by virtue of (1), (3) W x 0 = 0. Therefore, (4) 0 = 1. This in itself is a contradiction. Nevertheless, let us pursue the matter further. Let z be any number. Then by (2), we have (5) z = z x 1. Appealing to (4), plug in 0 in place of 1 in equality (5), we obtain z = z x 0, Therefore, z = 0. This proves that: If zero were to have a multiplicative inverse then there would exists only a single unique number. That is there is no number but zero. Whoever is ready to divide by zero must be living in a universe made up of a single atom. He or she is that only atom! (By, the way the above proof holds, not only for integers, but also for rational, real and complex numbers. In fact it applies for any field23.) 23

The author pleads guilty of smuggling a rich mathematical term with no further explanation!

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Incidentally, there is another misconception about zero that needs clarification. Many scientists confuse the limiting process: Limit (1/x) as x approaches 0 (which is alleged to be infinity), with division by zero. Consequently, they mistakenly claim that a number divided by zero is infinity. This is a two-fold misconception. To dispel the confusion let us clarify the meaning of the limiting process: The statement: Limit (1/x) as x approaches 0 is infinity means that: For any positive real number !, there is a positive integer N such that for every non-zero real number y, IF the absolute value of y < 1 / N THEN The absolute value of 1 / y > !.

Thus, the limiting process does not involve any division by zero at all. Incidentally, the statement written in the shaded box does not imply nor concede the existence of a number called infinity. Infinity is only a word that shortens the statement, written in the shaded box. It is not an element in the system of real numbers and has no relation with infinite sets!

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This needs a slightly higher bit of mathematical sophistication than the rest of the book.

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05.11. ZERO IS STILL A MISCHIEVOUS NUMBER

Zero is a troublemaker! It is still bringing trouble even in our modern time. Almost the population of the whole world commemorated the third millennium on January 1, 2000. If there should be a celebration on January 1, 2000, it must have been on the passing of 1999 years since the calendar was established. In reality, the new millennium started on January 1, 2001. The worldwide confusion resulted because there was no year numbered zero!

This is the end of our detour. We turn back to Mesopotamia to observe the people over there performing their arithmetic.

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06.

Arithmetic To manipulate = to control election results - Webster’s Dictionary

06.1. ARITHMETICAL TABLETS Advanced societies require a great many enterprises such as: Engaging in commerce. Digging canals. Building of embankments. Erecting of temples and buildings. Feedback from astronomical “exercises.” Setting of calendars.

In order to accomplish works such as these, not only the concept of numbers and their methods of representation are needed, but also numbers have to be added,

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subtracted, multiplied, divided, extracting their roots, â&#x20AC;Ś etc. Manipulations with numbers depend heavily on the way they are recorded. Mesopotamians did most of the arithmetical operations essentially in analogous manners to the ways we do arithmetical operations in modern times. No wonder! Both are place-value systems. However, the largeness of base-sixty has some influence. Tables for additions, multiplications are so large that ancient and modern students would have difficulty in memorizing. Therefore, Mesopotamians resorted to construct arithmetical tables for many purposes: for addition, multiplication, inverses, squares, cubes and exponential functions (which were used to calculate compound interest.) The tables of the Mesopotamians served their purposes, as calculators serve the purposes of modern society. The mathematicians of Mesopotamia demonstrated that they had amazing skills in those constructions and showed remarkable depth of mathematical understandings. Moreover, they utilized those tables to the paramount: in solving linear, quadratic, and even some cubic equations and simultaneous equations.

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06.2. MESOPOTAMIANS NUMERICAL ANALYSTS

WERE

Mesopotamians possessed a good sense of numerical approximations. They were - in our modern terms - numerical analysts. They utilized their sense of approximation to the utmost in their arithmetical manipulations. Doing mathematics for practical purposes is somewhat different from doing mathematics in school textbooks. In the latter, exercises are often, so to speak, â&#x20AC;&#x153;cannedâ&#x20AC;? problems; one gets accurate results. On the other hand, when doing mathematics for practical purposes, obtaining accurate results is rare. Hence, mathematicians (modern, as well as, ancient) pre-assign a tolerable margin of acceptable error. One would say that the calculation would be acceptable if performed up to so many decimal places (or sexagesimal places.) There are no excavated tablets claiming that the Mesopotamian mathematicians knew the following statements (expressed in modern notations):

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If – 1< a <1 and is negligibly small number. (That is a is “so small” that it lies below the pre-assigned acceptable tolerance.) Then:

(1)

(2)

1+ a ! 3 1+ a !

(3)

= app

1– a

= app

1+ a

! = app

1+ a

(1+ a)

2 3

! !

! Yet, it is plausible to conclude!based on their constructed mathematical tables that they did know them or they knew similar statements. Thus, the following proofs of the above three statements are immediate consequences of negligibility of a: We know that Hence, Therefore,

136

(1+a)(1–a) = 1 – a2 (1+a)(1– a) =app 1 1/(1 + a) =app 1–a.

The proofs require a slight refreshing of high school mathematics.

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In addition, since

(1+ a)2 = 1+ 2a + a 2 = app 1+ 2a Therefore ( 1+ a ) 2 2

1+ a =app

Consequently

1+ a

1+ a = app

! For the third statement, we know: 3 2 3 (1+ a) ! = 1+ 3a + 3a + a

= app 1+ 3a

Therefore

app

Hence,

= !

1+ a =

(1+ a 3)3 !

app

1+ a

Mesopotamians used the first of the above three statements mainly to calculate inverses and eventually to carry out long division, the second and the third to calculate the square roots and cube roots respectively and to solve different equations. Once they got a good approximation by using the above or analogous rules, they often applied the same rule to get a better approximation. Thus, they were almost within reach of the notion of convergent sequences.

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06.3. TABLES FOR RECIPROCALS OF NUMBERS Table 06.1 provides inverses of some sexagesimal “baby- numbers.” By simply utilizing the factors of the base b (which is sixty), it is easy to obtain the inverses of entries in the table. Here are some sample calculations of inverses. These calculations are intended to help the reader. We leave the rest to be checked by him or her. [2] x [30] = b, hence [1] / [2] = [30] / b. Therefore [1] / [2] = [0].[30], [3] x [20] = b, hence [1] / [3] = [20] / b. Therefore [1] / [3] = [0].[20], [4] x [15] = b, hence [1] / [4] = [15] / b. Therefore [1] / [4] = [0].[15], [1] / [8] = ([1] / [2]) x ([1] / [4]) = [0].[30] x [0].[15] = [0].[7][30].

A reader looking at the table of inverses observes the entries for 1 / 7, 1 / 11, 1 /13, … etc, are missing from the table. These are the “sexagesimally recurring fractions.” Mesopotamian mathematicians seemed to have a glimpse of the concept of recurring fractions. They would write down in their inverse tables an approximation to such numbers and

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add a remark such as: “The number does not divide.” They mean that at least one of factors of the number is not a factor of sixty. To refresh the memory of some readers, the following are examples of recurring fraction in the decimal system. 0.33333 …; 6.757575 …; 12.768653653653 … They are numbers represented by infinitely repeating numerals. The following set of numbers partially augments the mentioned table, where the subscript r stands for recurrence. 1 / 7 = [0]. ([8][34][17]) r, 1 / 11 = [0].[5]([27][16][21][49]) r 1 / 13 = [0]. ([8][12][5]) r, 1 / 14 = [0].([4][17][8][34]) r 1 / 59 = [0]. ([1]) r , 1 / 61 = [0].([0][59]) r. A question directed to the reader: Do the last two numbers ring a bell 24? As a further illustration, let us find an approximate inverse of a number that does not appear in the table, say, [29]. One finds in the table that the nearest number to [29] having an inverse is [27]. Its inverse is [0].[2][13][20]. 24

1/9 = 0.(1) r

and 1/11 = 0.(09)r

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Remember our convention that a numeral inside a square bracket stands for a baby-number in sexagesimal system

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Therefore, an approximate inverse of [29] is [0].[2][13][20]. We conjecture that Mesopotamian mathematicians, in order to obtain a better approximation for the inverse of [29], would probably proceed along the following path: [1] / [29] = [1] / ([27] +[2]) = [1] /([27] x ([1] + [2] / [27])) = [1] / ([27x (1 + z)) where, z stands for [2] / [27]. Assuming that z is negligible [1] / [29] = [1] / ([1] +z). Then appealing to (1) in the above statements, the Mesopotamian mathematicians would get: [1] / [29] = ([1] / ([27]) x ([1] â&#x20AC;&#x201C; a) = ([0].[2][13][20]) x ( [1] â&#x20AC;&#x201C; [2] x ([0].[2][13][20])) = ([0].[2][13][20]) x (1â&#x20AC;&#x201C; [0].[4][26][40]) = ([0].[2][13][20]) x ([0].[55][33][20]) = app [0].[2][3][11]. If a better approximation was required then the process had to be repeated.

140

This means that the approximation is correct to one sexagesimal place. Why?

From the table [1] / [27]= [0].[2][13][20])

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As another illustration, let us find an approximate inverse of [31][24]. [1] / [31][24] = [1] / ([30][0] + [1][24]) = [1] / ([30][0] x ([1] + [1][24] / [30][0])) = [1] / ([30][0] x ([1] + c)) Remember our convention that = app ([1] / ([30][0]) x ([1] – a numeral c)) inside a square bracket stands = [0].[0][2] x ([1] – ([1][24] x for a baby[0].[0][2]) number in = [0].[0][2] x ([1] – sexagesimal system. [0].[2][48]) = [0].[0][2] x [0].[57][12] = [0].[0][1][54][24].

141

Denote [1][24] / [30][0] by c

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1/2 = 30 / b = [ 0].[30] 1 / 3 = 20 / b = [ 0 ].[20]

1 / 4 = 15 / b = [0].[15]

1/5 = 12 / b = [0].[12]

1 / 6 = 6 / b = [0].[10]

1 / 8 = 7 / b + 30 / b2 = [0].[7][30]

1 / 9 = 6 / b + 40 / b2= [0].[6][40]

1 / 10 = 6 / b = [0] . [6]

1 / 12 = 5 / b = [0].[5]

1 / 15 = 4 / b = [0].[4]

1 / 16 = 7 / b + 45 / b2 = [0].[3][45]

1 / 18 = 2 / b + 20 / b 2 = [0].[3][20]

1 / 20 = 3 / b = [0].[3]

1 / 24 = 2 / b + 30 / b 2 = [0] [2][30]

1 / 25 = 2 / b + 24 / b2 = [0].[2][24]

1/27=2/b)+(13/b2)+(20/b3) =[0].[2][13][20]

1 / 30 = 2 / b = [0].[2]

1/32 = 1 / b + 52 / b2 + 30 / b3

1 / 36 = 1 / b + 40 / b2 = [0].[1][40]

= [0].[1] [52][30] 1 / 40 = 1 / b + 30 / b 2 = [0].[1][30]

1 / 45 = 1 / b + 20 / b2 = [0].[1][20]

1 / 48 = 1 / b + 15 / b2 = [0].[1][15[

1 / 50 = 1 / b + 12 / b2 = [0].[1][12]

1/54=1/b+6/ b2 + 40 /b3 = [0].[1][6][40]

1/60 = [0].[1]

Table 06.1.

This is a table of inverses for sexagesimally “baby numbers” “that” divide. The letter b stands for the base sixty.

06.4. ADDITIONS AND SUBTRACTION

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Mesopotamians might have constructed their tables of additions by starting with one and adding 1 to each preceding number. They performed the processes of addition and subtraction in analogous manners to the ways we usually do them in our familiar decimal system. We give in the following figure illustrative examples. The reader is requested to redo the calculations and to check the calculations by transforming all the numbers into the decimal system Remember our convention that a numeral inside a square bracket stands for a baby-number in sexagesimal system

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–

Find the sum of:

Find the difference between:

[53][17][24][38] and [9][3][44][7]

[1]

[17][3][41][2] and [12][51][15]

[1] [1]

[1][0] [1][0] [16]

[33][53][17][24][38] [9] [3] [44] [7]

Table 06.2.

[2]

[16] [50] [49] [47] Where “the small numerals” are the numbers after borrowing.

Illustrative examples for the additions and subtraction operations in sexagesimal system

06.5. MULTIPLICATION AND DIVISION It was difficult for Mesopotamians to memorize the multiplication table in the sexagesimal system. It is too large to be memorized. It has 3600 entries. People whose jobs involve performing multiplications have to have a copy of the table of multiplication at his

144

[1][0]

[12] [51] [15]

[34] [2] [21] [8] [45] Where “the small numerals” are the “in hand”.

[2] [40]

[17] [3] [41]

- 145 -

or her desk, just as their modern counterparts have with our calculator. The reader can find at the end of the chapter a necessarily an incomplete table of multiplication in the sexagesimal system; there are wide gaps in the table. The interested reader is urged to complete at least some of the gaps as an â&#x20AC;&#x153;entertaining exerciseâ&#x20AC;? in sexagesimal system. Mesopotamians must have performed the operation of multiplication in a procedure similar to our modern procedure with the decimal system. Figure 06.3 presents an example illustrating the procedure. Mesopotamians did not seem to possess a method for performing long division. The reason is likely to be in the difficulty in memorizing multiplication table of the large sixty-based numeration system. They carried division through multiplying by the inverse with the aid of table of inverses. For example: [1][5] / [16] = [1][5] x [0].[3][45] = [4].[3][45]. Another example: [1][5] / [7] = [1][5] x [0].[8][32][7] r =app [1][5] x [0].[8][32][7] = [9].[14][8][35]. A better approximation [1][5] / [7] = [1][5] x [0].[8][32][7] [8][32][7]

145

Remember our convention that a numeral inside a square bracket stands for a babynumber in sexagesimal system

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= [9].[14][57][44]. This is correct to the third sexagesimal place.

Multiply

[2]

[55]

[19]

[4]

[3] [1]

[19]

[17]

[27] [6]

[25] [57] [48] [16]

[1] [12]

[16]

[14]

[16]

Answer

[1]

[24] [1]

[2] [8] [2]

[57] [22

[6] [45] [3]

[1]

[38] [45] [33] [4]

Table 06.3. An example of the procedure of multiplication in sexagesimal system. The “small numerals” are self-explanatory.

06.6. SQUARES NUMBERS

AND

CUBES

Two tablets were found dated about 2000 BC. They give squares up to 59 and cubes up to 32. The method of construction of

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the table of squares must have been depended on the formula (n + 1)2 = n 2 + n + n + 1, which is easily suggested when considering objects arranged in a square as shown in the Table 06.4: Thus, they obtained the following: [1] [2] = [1], [2] [2] = ( [1] + [1]) [2] = [1] [2] +[1] +[1] + [1] = [1] + [2] + [1] = [4], [3] [2] = ([2] +[1]) [2] = [2] [2] + [2] +[2] + [1] = [4] + [4] + [1] = [9], … ………… [7] [2] = ([6] + [1])[2] = [6] [2] + [6] + [6] + [1] = [36] + [12] + 1 = [49], [8] [2] = ([7] + [1]) [2] = [7] [2] + [7] + [7] + [1] = [49] + [14] +[1] = [1] [4], [9] [2] = ([8] + [1]) [2] = [8] [2] +[8] + [8] + [1] = [1] [4] + [8] + [8] +[1] =[1][21] … … … …

147

Remember our convention that a numeral inside a square bracket stands for a babynumber in sexagesimal system

- 148 -

Figure 06.4

148

Objects arranged in a square. (n+1) 2 = n2 + n + n + 1 = n2 + 2 n + 1

- 149 -

Sometimes, in order to check their multiplications, Mesopotamians utilized their tables for squares and employed the formula: a x b = ((a + b) 2 â&#x20AC;&#x201C; (a â&#x20AC;&#x201C; b) 2 ) / 4. Does this method cross the minds of our modern high school students? For obtaining the cubes of numbers a similar procedure was utilized by employing the formula (n+1) 3 = n 3 + n 2 + n 2 + n2 + n + n + n + 1 = n 3 + 3n 2 + 3n + 1 This can be obtained using a pile of objects in a cube.

06.7. SQUARE ROOTS

ROOTS

AND

CUBE

In CHAPTER 08, there will be a brief discussion about a small sized tablet labeled YBC7289. The tablet has a picture of a square with its two diagonals. There are two numbers along one of the diagonals of the square. Here is a picture of the tablet.

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Figure 06.5. Yale YBC 7289 2CYBC7289

As usual with the Mesopotamians expressions of numbers, the tablet does not have any indications on how the numbers were obtained. As usual, with Mesopotamian tablets, the inscribed numeral is [1][24][51][10] and has no sexagesimal point. The geometric configuration suggests that the number might be somewhat related to [2] . The calculations that follow show that it is. In fact, it is a â&#x20AC;&#x153;goodâ&#x20AC;? approximation to [2] . Here is a possible method that the ! Mesopotamians might have used in order to obtain that number as successive ! approximations. Calculating the square of [1].[30] we find that it equals [2].[15]. Hence, [1].[15] is a first approximation to [2] . To find a second approximation:

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[2] = =

= !

! = !

([1].[30])2 (1" .[15] [2].[15])

([1].[30]) 1" .[15] [2].[15] NOTE [1] [2].[15] = [1] ([2])([1]+ .[15] [2]) = [1] ([2])([1]+ .[7][30])

=app

Remember our convention that a numeral inside a square bracket stands for a baby-number in sexagesimal system

[2].[15] " .[15] ([2].[15])(1" .[15] [2].[15])

By appealing to statement (1) we obtain:

([1].[30]) 1" (.[15])(.[26][15]) ! !

Plug this result in the above equality, we obtain the equality on the left !of the box.

! = =app = ! = ! = ! !

=app (.[30])([1] " .[7][30]) = (.[30])(.[52][30]) = .[26][15]

([1].[30]) 1" .[6][33][45] ([1].[30])(1" .[6][33][45] [2])

! By appealing to statement (2) we obtain the statement on the ! left of the box

([1].[30])(1" .[3][16][52][30] ([1].[30])(1" .[56][43][7][30] [1].[25][4][42]

How good is this approximation? It does not seem to be a â&#x20AC;&#x153;goodâ&#x20AC;? enough. Assuming [1].[25] as a second approximation, let us find a third approximation.

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[2] =

= !

([1].[25])2 (1" (.[0][25]

([1].[25])( 1" (.[0][25]

[2].[0][25]

)

))

NOTE: ([1] ) = [1] ([2])([1]+ (.[0][25] )) [2].[0][25] 2

! !

=app ([30])(1" .[0][12][30]) = ([30])(.[59][47][30]) = .[29][53][45] . Plug this result in the preceding equality on the left of the box.

= =app

! ! ! !

By appealing to statement (1) we obtain

([1].[25])( 1" (.[0][25])(.[29][53][45])) !

Remember our convention that a numeral inside a square bracket stands for a baby-number in sexagesimal system

([1].[25])2 " .[0][25]

= = =app

! ([1].[25])( 1" (.[0][12])[28][27][24])) ! ([1].[25])(1" (.[0][12][28][27][24]! )) 2 ([1].[25])(1" .[0][6][14][13][42]) ([1].[25])(.[59][53][45][46][8]) [1].[24][51][57]

Appealing to Statement (2)

Therefore, the inscribed number [1].[24][51][10] is a third approximation to 2 . It corresponds to 1.41421296 in decimal notation. It is accurate to five decimal places, which is more than necessary for most practical purposes. This was! a lengthy

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- 153 -

calculation. Mesopotamians must enjoyed doing lengthy calculations.

have

There is no need to burden the reader with similar lengthy calculations for finding cube roots.

153

- 154 Remember our convention that a numeral inside a square bracket stands for a baby-number in sexagesimal system

[1] [2] [3] [4] [5] [6]

[2] [4] [6] [8] [10] [12]

B [30] [1][0] [31] [1][2] B R E [56] [1][52] [57] [1][54] [58] [1][56] [59] [1][58] [1][0] [2][0]

[3] [6] [9] [12] [20] [18] R [1][30] [1][33] A K [2][48] [2][51] [2][54] [2][57] [3][0] Table 06.6.

B R E A K B R E A K

[30] [1][0] [1][30] [2][0] [2][30] [3][0] E [15][0] [15][39] BREAK [28][0] [28][30] [29][0] [29] [30][0]

B R E A K

[58] [1][56] [2][54] [3][52] [4][50] [5][48] A [29][0] [29][30] B R [54][8] [55][6] [56][4] [57][2] [58][0]

An incomplete multiplication table in the sexagesimal system.

The reader is requested to fill the gaps of 3576 entries as an entertaining exercise!

154

B R E A K

[59] [1][[58] [2][57] [3][56] [4][55] [5][54] [29][30] [30][29] E A [55][4] [56][3] [57][2] [58][1] [59][0]

[1][0] [2][0] [3][0] [4][0] [5][0] [6][0] K [30][0] [31][0] K [56][0] [57][0] [58][0] [59][0] [1][0][0]

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0.7 Algebra Algebra â&#x20AC;Śfrom Arabic al-jabr - Encarta World Dictionary

07.1. IMPORTANCE OF ABSTRACTION Many Mesopotamian tablets were found to be concerned with the digging of canals. Take for example the statement found in an excavated tablet: â&#x20AC;Ś Beyond the ditch I made a dike, one cubit per cubit is the incline this dike. What are the base, the top and the height of it? And that is the circumference?

Canals were important to the Mesopotamian administrators. Inland waterways were vital in both irrigation and transportations for civil, as well as for military purposes. For each excavation project, the administrations must have asked the mathematicians toward calculate the volume of the earth to be excavated, to estimate the number of workers needed to do the work in a certain prescribed period and to determine the wages of workers.

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This should have recurred for several centuries. Mesopotamian mathematicians must have become experts in this field of knowledge and must have prepared tables ready to answer questions the government raised. A discipline, practiced for several centuries by specialists whose task is not only to apply it, but also to convey its methods and secrets to young students will eventually be studied for its own sake. This would ultimately lead to an abstraction. Thus, abstraction must take place as long as there are mathematically inclined minds. Algebra must eventually evolve from arithmetic. Since algebra allows better methods of calculations. Moreover, it is a natural outgrowth of a discipline studied and cultivated in schools. It is an apt time to make a short digression into the importance of abstraction in general, and in mathematics in particular. Abstraction is the procedure of mentally stripping things out of a particular situation. For example, consider the table I am sitting at this moment. The table is made up of a certain material, of a certain thickness, in a certain shape, of a certain height, standing on

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a certain number of legs, to be used for a certain purpose, … etc. Now if you strip one or more of these properties from the table, then you can talk, think, imagine many other sorts of tables. You may order a table made of glass, or a table for dining purposes, or table for playing ping-pong. You may even imagine a table that never existed and no one ever imagined before. Abstraction is one of the ingredients of thought. Without it, a result obtained to deal with a certain situation, would apply to that situation and only to that situation. With the procedure of abstraction, one might find new results, which more often than not are applicable to further situations or cases. Abstraction is one of the characteristic properties of mathematics. In fact, mathematics is the most abstract field of knowledge. It is this characteristic property that explains the “Unreasonable effectiveness of Mathematics” 25 and that makes mathematics more applicable (in spite of the wide spread opposite opinion). An illustration will clarify the above assertion.

The phrase is borrowed from the title of E. P. Wigner’s 1960 paper.

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There are numerous mathematical possible illustrations but, in order to avoid paddling into deep mathematical waters, we choose for our illustration a simple and almost trivial example. Suppose it was found, after a “long research”, that the union of two trees with three trees is the same as the union of three trees with two trees. This hypothetical research was done on trees and using properties of trees. The result thus obtained, applies for trees only. It does not necessarily apply for men, women, houses, apples, or oxen. When a mathematician looks into the matter, he mentally strips the trees out the result, and ignores their roles. That is, he makes an abstraction. Thus obtaining: 2+3 =3+2 This statement applies is not only true for trees but also for men, women, apples, houses, oxen, weights, areas, and volumes …etc; in fact, it applies for many things. Furthermore, the statement ‘2 + 3 = 3 + 2’ seems initially to be true for the particular pair of numbers 2 and 3. It might not be true for other pairs of numbers. Again, a mathematician studying the latter equality strips the particular numbers 2 and 3 out from that result, and ignores their roles. Thus, he obtains the even more abstract statement:

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x + y = y + x. This new result is even more applicable. It is not only true for the particular pair of numbers 2 and 3 but for the pair 257, 378, and other pairs. In fact, it is true for ANY pair of integers. Further stripping the above formula from being integers out the above statement, one obtains the same formula: ‘x + y = y + x’ to be true for rational numbers, real numbers, and complex numbers too. The new result is so important that it is endowed with a name: we say, the binary operation of addition is commutative. Mathematicians are tempted to look for other commutative binary operations. They find many others, for example, multiplication of numbers. Thus, mathematicians discover a somewhat analogous structure relating the two operations. Some mathematicians might dare to look for an operation, which is not commutative. They soon find an instant of a non-commutative operation. It is exponentiation; 23 is not the same as 32. More mentally courageous mathematicians (usually labeled ‘pure mathematicians’) begin to probe noncommutative algebra, and to their own surprise, and amazements of other scientists they find that non-commutative algebra has many applications in the fields of physics and

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chemistry, as well as in mathematics. Thus, the boundaries that separated pure from applied fields become blurred. Thus, in short, the characteristic property of abstraction renders mathematics more applicable and eventually more useful in exploring nature. Classical algebra is some sort of an abstraction, as well as, a generalization of the methods of arithmetic. In the discipline of algebra, problems are classified into different classes. The study of algebra provides methods that are appropriate to deal with, for each particular class. One way of classification can be made according to the number of variables, another way of classification according to the highest degree of the variables involved, â&#x20AC;Ś etc. Mesopotamian mathematicians dealt with equations of first, second and third degree and with equations with two or more unknowns. In their dealings and handling equations of degrees higher than the first, they had exhibited the possession of extraordinary manipulative dexterities, maturities and flexibilities in algebraic skills. Algebra, in our modern society, deals with variables denoted usually by alphabetic letters standing for numbers. However, the

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algebra of the Mesopotamians was rhetoric. It is easy to understand the problems that they raised, but without the already proper known machinery, it would be difficult to follow their procedures for solutions.

In the rest of the chapter, we will illustrate the methods employed in dealings with different algebraic problems by the Mesopotamian mathematicians. We would like to draw the attention of the reader to the fact that this chapter requires from him or her, a clear memory of high school mathematics. The reader who does not posses that requirement may quickly skim over the ‘equation-formula-solution’ part of the exposition (but not the comments in between) without losing continuity with the rest of the chapters of the book.

07.2. SOLUTIONS EQUATIONS

LINEAR

The simplest equation to solve is the linear equation in one unknown. It is of the form a x = b, where a is not zero. He or she starts to solve the equation, by consulting the table of inverses to find 1/a and then multiply the obtained inverse with b. An example of such a problem is found in a tablet in the following form:

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If a=0 and b≠0 then there is no solution. If a=0 and b=0 then every real number is a solution.

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[0].[40] of [0].[40] of a certain quantity of barley is taken, [1][40] units of barley is added and the quantity is recovered. Find the quantity of barley.

The equation to be solved, using our modern notation (but with sexagesimal numeration system written in square brackets) is: ([0].[40] x [0].[40]) z + [1][40] = z The method of solution is the standard method: [0].[26][40] z + [1][40] = z Hence, ([1] â&#x20AC;&#x201C; [0].[26][40]) z = [1][40] [0].[33][20] z = [1][40] z = [1][40] / [0].[33][20] = [1][40] x [1].[48] = [3][0] The Mesopotamians would provide the same steps. The only difference is that they use words. They would write something like the following (with no sexagesimal point). Multiply [0].[40] by [0].[40] you obtain [0].[26][40]. Subtract [0].[26][40] from [1], you get [0].[33][20]. From the table of inverses find the inverse of [0].[33][20], you obtain [1].[48]. Multiply [1][40] by [1].[48], you get the answer [3][0]

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(The Mesopotamians would have written simply [3].) The class of linear equations with a single variable is the simplest possible class of equations; we will consider in the following sections the less simple classes that Mesopotamians tackled.

07.3. SOLUTION EQUATIONS

QUADRATIC

One should remember that the Mesopotamians used to allow only positive solutions. In fact, concrete situations, which led to these equations, had to have positive solutions 26. The Mesopotamians, not possessing negative numbers, had to classify quadratic equations into two classes in the following forms: x2 + bx = c

and

x 2 = c + bx.

Negative numbers did not appear until the 16 th Century AD.

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Where, b and c are positive numbers. In our modern societies, that concedes the existence of negative numbers these two classes confluence into a single class. Obviously, in the second case, the solution has to be greater than b. Mesopotamian mathematicians used several methods in attacking quadratic equations. One of the methods they used was based on the standard procedure of completing the square. The method is shown in the following table: Adding 2

( b 2 )2

to both sides of each of the equations one gets:

x + bx + (b 2) = c + (b 2)2

x 2 + (b 2)2 = c + bx + (b 2)2

These yield

! !

(x + b 2)2 = c + (b 2)2 ! ! x = c + (b c)2 " (b 2)

(x " b 2)2 = c + (b 2)2 Therefore

x = c + (b c)2 + (b 2)

These are the solutions of the two equations respectively

! Let us have a concrete ! example from a tablet, which states the following problem: The area of a rectangle is [1][0] and its length exceeds its breadth by [7]. What is the length and the breadth of the rectangle?

The problem leads to the equation

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x (x + [7]) = [1][0] The standard method of solution would lead (using sexagesimal numeration) to x=

([7] [2])2 +[1][0] "[7] [2]

! !

= ([3].[30])2 +[1][0] "[3].[30] =

[12].[30]+[1][0] "[3].[30] = [1][12].[30] "[3][30] = [8].[30] "[3].[30] = [5]. The Mesopotamian mathematicians ! wrote on tablet the solution in rhetoric form in ! way: the following

Half of [7] is [3].[30]. Square [3].[30]. We get [12].[25] Add [12].[15] to [1][0] to get [1][12].[15] Use the table of squares, you find that square of [8]. [30] equals [1][12].[15]. Subtract [3]. [30] From [8]. [30]. To give the answer [5].

This rhetorical method reminds us of computer program in one of computer languages: Input b: Input c B1 =b/2 B2 = (B1) 2 A = B2 + c S =â&#x2C6;&#x161;A

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R = S + B1 Print “solution =”; R

The Mesopotamian mathematicians seemed to explain their methods by illustrating their procedures through recording an illustrative example. Their procedures are what we, today, call the - algorithms or the programs for the solutions. This seemed the way to “publish” their findings. The illustrative examples were usually obtained from a concrete situation.

07.4. ALTERNATIVE METHOD FOR SOLUTION OF QUADRATIC EQUATIONS Completing the squares was not the only method the Mesopotamians used to solve quadratic equations. Reducing the quadratic to simultaneous bilinear equations was another method. The best way to explain it, is by a concrete example: I need six times the side of a square the result is [1][41][15].

In modern terminology and using decimal system, the problem can be expressed in the form:

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[1][41][15]=6075

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x2 + 6 x = 6075 They would have written, using our modern notations with decimal numeration, in the form: x (x + 6) = 6075 Denote x + 6 by y. Thus, we have the following bilinear simultaneous equation y â&#x20AC;&#x201C; x = 6 and xy = 6075. Squaring both sides of the first equation and multiplying both sides of the second equation by 4 we obtain: (y â&#x20AC;&#x201C; x) 2 = 36 and 4 x y = 24300 Adding the two equations yields: (y + x) 2 = 24336 = 156 2 Therefore x + y = 156 and 2x = 156 â&#x20AC;&#x201C; 6 = 150 Finally, the solution is: x = 75 These are ingenious methods for attacking problems. The readers might be silently demanding explanations of how those ancient people obtained such imaginative methods. Our answer is disappointing. We do not know for sure. None of the Mesopotamian tablets discovered until now gave any indication on the method they used to obtain those results. We can only emphasize that those methods for solving equations point out that the Mesopotamian mathematicians possessed profound mathematical understandings and skills, whether or not their methods satisfy the Euclidean criteria of proof. We will have elaborate discussions about these matters in CHAPTER10.

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The Mesopotamians even tackled the cubic equation. Next section will describe their novel method.

07.5. SOLUTION OF CUBIC EQUATIONS The Mesopotamian mathematicians were almost addicted to constructing tables. Surely, those tables were not made for fun. They had 3 2 constructed tables for: n + n , where n is an integer. With the aid of these tables, they were able to solve some cubic equations. For example: ax 3 + bx 2 = c, where neither a nor b is zero

The method they employed in tackling such a problem is by multiplying both sides by a2 / b3. Thus, obtaining the equation: (ax / b)3 +(ax / b) 2 = ca 2/b3. The left hand side of this equation is of the form n3 + n2. Then appealing to the table of n3 + n2, they could read the number in the table closest to ca2/b3 from which they would calculate an approximate solution by multiplying by b/a. A better approximation could be obtained through interpolation. Some mathematically inclined readers might point out the cubic equation the Mesopotamians solved is not a general cubic

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NOTE: If either a, or b were zero then the general equation would degenerat e into a simple quadratic or a simple cubic.

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x3 + Ax2 + Bx + C = 0. The linear term Bx is missing from the equation they solved. This objection though true is irrelevant. Why? The reason is that the cubic equation: ax3 + bx2 = c, in spite of the missing linear term, is in a certain sense, the most general cubic equation. In fact, there is a standard method of reducing the general cubic form z3 + A z2 + B z + C into the above form by “killing” the term Bz. The method is accomplishable by a transformation:27 z = w + h. This transformation leads to a certain quadratic form in h. The number h can be so chosen to make that quadratic form to be zero. This means that h is to be chosen as the root of a quadratic equation. (The details can be found in the footnote.) Did the Mesopotamian mathematicians know all about this method of reduction? 27

Consider the transformation z = w + h for some number h to be evaluated in the sequel. Plug the quantity w + h in place of z in z 3 + A z 2 + B z + C. The following is obtained: (w + h) 3 +A(w +h) 2 + B(w + h) + C. = w3 +(3h + A) w2 + (3 h2+2 A h + B ) w + ( h3 + A h2 + B h + C) In order to “kill” the linear term, choose h so that 3h2 +2Ah +B = 0. Thus we have w3 + A’ w2 + C’ for some A’ and C’.

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This is an interesting question. No one knows the answer for sure. However, we add parenthetically that we believe that the reduction process is within the capabilities of the Mesopotamians. Nonetheless, the mere fact that the Mesopotamian mathematicians had constructed tables for n3 + n2 and NOT for n3 + n2 + n makes it more plausible that they knew what they were doing.

07.6. SOLUTIONS OF SIMULTANEOUS EQUATIONS The method used by Mesopotamians in attacking simultaneous equations, is the standard method except they used it in a rhetoric manner. Let us explain by an example from a tablet. Calculate the length and the width of a rectangle whose perimeter is 54 and the sum of its length with a third of its width is 19.

This problem leads to the simultaneous equations: X + Y / 3 = 19 2X + 2Y = 54. The length and width are represented by X and Y of the rectangle respectively. The standard way to solve this system of

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equations is to multiply the first equation by 6. Thus obtaining: 6X + 2Y = 114 2X+ 2Y = 54. “Kill” Y by subtracting the second equation from the first. Thus, we get 4X= 60 Therefore X=15 and Y=12. The Mesopotamians would have the same ideas except expressed by words. They would proceed as follows: 19 x 6 =114 114 – 54= 60. 60 x 1/4 =15 (the length); 54 x 1/2–15 =12 (the width.)

07.7. ELIMINATION METHOD FOR SOLVING SIMULTANEOUS EQUATIONS

The mathematicians of Mesopotamia used an ingenious method for solving simultaneous equations. Let us explain the method through a numerical example: Solve the system of equations x + 3y =1000 2x – y = 600. Start by making a guess on the value of x, say, 250. Knowing this is only a guess set a correcting term d.

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x = 250 + d. Consequently, the second equation becomes y = 2x – 600 = 2(250 +d) – 600 = 2d – 100. Plug in the values of x and y in the first equation we obtain an equation in d: (250 + d) + 3(2d – 100) =1000 7 d = 1050. Therefore Hence d = 150. From which we conclude that: x = 250 +150 = 400 and y = 2(150) –100 = 200. This method can be used when one equation or even both are not linear. No tablet has been found using this method of attack when the equations are not linear.

07.8. QUADRATIC EQUATIONS

SIMULTANEOUS

A tablet was found in Tell Dhibayi near Baghdad, dated between 1900 BC –1600 BC. The word “Tell” means hill in the Arabic language. The tablet contains a problem that needs knowledge of Pythagorean theorem and led to a pair of quadratic equations. The tablet gave the solution in the rhetoric method.

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The Mesopotamian mathematicians artfully sidestepped the difficulty by transforming them into a pair of linear equations. The problem in Tell Dhibayi tablet is: Given the area of a rectangle is [0].[45] and the length of its diagonal is [1].[15]. Find the sides of the rectangle. This problem leads to a quadratic simultaneous equation. In modern notations (with sexagesimal numeration system): [2] [2] [2] xy = [0].[45] and x + y = ([1].[15]) These can be written [2] xy = [1].[30] and x [2] +y [2] = [1].[33][45], The method employed in the tablet for finding the solution, runs like the following: Adding and subtracting one obtains: x[2] +[2] xy +y [2] = [3].[3][45] and x [2] – [2] xy + y [2] = [0].[3][45] These can be rewritten in the forms [2] [2] (x + y) = ([1].[45]) and (x –y)[2] = ([0].[15]) [2] For each equation, we take square roots of both sides, we obtain: x + y = [1].[45] and x – y = [0].[15] Therefore, x = [1] and [y] = [0].[45].

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07.9. FINITE SEQUENCES

Inscribed on a tablet, dated about 300 BC, two interesting finite sequences with their sums as follows (in modern notations): (1) 1 + 2 + 22 + … + 29 = 29 + 29 –1, (2) 12+22+32+ … +102 = (1 x (1/3) + 10 x (2/3) ) = 385.

55.

In this section, we speculate about whether or not the Mesopotamian mathematicians knew the general formulae for these finite sequences and the possible methods they might have obtained them if they did. Mesopotamian mathematicians having reached such height of sophisticated algebraic knowledge must have known the general formula for the (easier) summation of the finite sequence: (3) 2 (1 + 2 +3 + … + n) to be n (n + 1).

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They could have obtained the above result by rewriting the finite sequence (3) in the form: 1 + 2 + 3 +…+n + n +(n –1) +(n – 2) + … + 1. Furthermore, by adding each pair of numbers whose numerals are one “above” the other, it would be easily observed that the sum of each pair is n +1. The fact, that there are number of such pairs is n, leads to (3) is true. We conjecture that Mesopotamians have known the sum of the finite sequence (2). Upon this conjecture, we can bet that they also knew the general formula: (4) 1 + 2 + 22 + … + 2n = 2n + 2n – 1. of which equality (2) is a special case when n = 9. They could have reached the result by observing that 1 + 2 + 22 + … + 2n = 1 x (1 + 2 + 22 + … + 2n –1 + 2n) = (2 –1) x (1 + 2 + 22 + … + 2n –1 + 2 n) = (2 + 22 + … + 2n) + 2 x 2n – 1 – (2 + 22 + … + 2n) By adding, one easily observes that the only surviving terms would be 2 x 2n and – 1. This proves equality (2) Upon the assumption that the Mesopotamians knew the finite sequence (4), let us consider how the Mesopotamians could have handled the sequence (3) of the sum of the squares.

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Performing multiplication

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The Mesopotamians, having constructed tables of cubes of numbers, should be aware of the fact that (k + 1) 3 = k 3 + 3 k2 + 3 k + 1. From which the following identity could be obtained: (k + 1) 3 – k3 = 3 k2 + 3 k + 1. In addition, very likely they experimented with special cases such as: 23 – 1 3 = 3 x 1 2 + 3 x 1 + 1 33 – 23 = 3 x 22 + 3 x 2 + 1 43 – 3 3 = 3 x 3 2 + 3 x 3 + 1 ………… 3 3 2 (n + 1) – n = 3 x n + 3 x n + 1. There is a great temptation for a mathematically talented person to add up the terms. Upon addition, the only surviving terms on the left hand side are (n +1)3 and – 1. The sum on the right hand side would be: 2 2 2 2 3 x (1 + 2 + 3 + … + n ) + 3 x (1 + 2 + 3 + … + n) + n, From which the one can conclude that: 2 2 2 2 3 3 (1 + 2 + 3 + … + n ) = ((n +1) – 1) – (3 n (n + 1) / 2) – n = n (n + 1) (2 n + 1) / 2. Therefore,

1 2n n(n +1) )( ). 3 3 2

(5) 12 + 22 + 3 2 + … + n2 = ( +

Formula (2) is a special case of (5) when n = 10. Though we! admit that there is a possibility that the author of the book is projecting himself or his ideas on the

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contributions of the Mesopotamians, yet we DO believe that there is nothing in the above arguments that the Mesopotamians could not handle. Whether they did or not is left for future excavations and investigations.

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08. Geometry Geometry is man made. Clouds are not spheres. Mountains are not cones, Nor does lighting travel in straight lines. Benoit Mandelbrot (Fractal Geometry of Nature)

08.1. INTRODUCTION It is true that man made geometry. Mother Nature â&#x20AC;&#x201C; as pointed out by Benoit Mandelbrot â&#x20AC;&#x201C; does not seem to like the geometry made by man. However, there is at least one exception. Light is an exception. It travels in straight lines28.

More precisely, light travels in Geodesic, which is the shortest path in a medium.

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Nature has contributed straightness to the human mind via the path traveled by light. Ancient and modern people had built two different kinds of structures: material and thought structures. Both kinds were great and magnificent. Both were built in order to understand Nature and eventually to control it. The Egyptian Pyramids are considered one of the seven wonders of the Ancient world. People in our modern age still wonder how was it possible to construct such huge monuments a couple of thousands of years ago? Mesopotamians had somewhat similar huge structures. Among which are the hanging gardens, made famous through the reports of Greek historians-tourists. Mesopotamians were also famous for the many other structures they had constructed which were called Ziggurats. The following is a sketch of the famous Ur Ziggurat.

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Figure 08.1

A sketch of ziggurat of Ur

Ur is an ancient town in south of Iraq situated on the Euphrates. It was the capital city of one of Sumerian governments. It is mentioned in the Bible as the birthplace of the legendary figure of the Prophet Abraham. The gigantic structure is impressive. Today, with all our sophisticated instruments in exact measurements, it is considered a significant occasion when lines driven through

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a mountain meet and make a tunnel. How much more wonderful, positioned on the corners of a rectangular region on the ground, with only the most elementary instruments at hand, to erect such a huge and elaborate structure? Does it not strike the reader as marvelous, with no transit instruments, to raise the ground platform some hundred and fifty feet in order to build the temple? Those ancient people must have possessed “sophisticated instruments of thought” that made them able to accomplish such a feat. They must have had a good amount of geometric knowledge as well as trigonometry. For some modern readers, those “sophisticated instruments of thought”, which are geometry, and trigonometry were boring subjects. Certainly, they are boring if badly taught. More often than not, they are presented to students as finished, dehumanized, and dry subjects with no motivations. The two subjects are “projected” onto modern students only as to satisfy requirements and pass examinations. However, gazing with admiration on the pyramids or the ziggurats and wondering about their almost miraculous construction might create some motivational appreciation.

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08.2. WHAT WERE ZIGGURATS FOR? Ziggurats were to the Mesopotamians as malls are to the Americans. Cities of modern societies compete with each other by the architectural beauty of their malls. Likewise, the cities in Mesopotamia also used to compete with each other by the design, beauty, and height of their Ziggurats. Ziggurats were religious centers. In fact, they were more than that. Those structures were also focal points of civic pride. Each city in Mesopotamia rose up around the shrine of a local city-god, as a reflection of the cityâ&#x20AC;&#x2122;s wealth and devotion to its god. The temples became elaborate structures. They were raised on a spacious platform nearer to god in the sky. Their high level was reached by a broad staircase, which may reach 150 feet high and was about 30 feet wide. The whole structure was called ziggurat, which meant holy-mountain. The temple grounds were quarters for priests, officials, accountants, musicians, singers, and workshops for bakers, pottery makers, leather workers, and jewelers. There were also enclosures for keeping the sheep and goats

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that were destined to be sacrificed to the temple god.29

08.3. NEED OF GEOMETRIC KNOWLEDGE The construction of ziggurats, temples, and other buildings call for: Calculations of volumes of bricks or stones of different shapes. Construction of straight lines on the ground and in space too. Construction of angles especially right angles on the ground and in space too. Construction of angles especially right angles in space. Construction of trigonometric ratios.

Similar needs come from other channels too. The Mesopotamians in order to irrigate their lands had to construct long canals in which water was to be diverted. This

Similar structures to ziggurats have been discovered in Central America for worshipping their gods.

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necessitated calculation of volumes of the earth to be excavated. Thus, the Mesopotamian mathematicians had to know the areas of simple figures and volumes of some solids. In fact, Mesopotamians were familiar with areas and properties of: Squares, rectangles, right angle triangles, isosceles triangles, parallelogram, and trapezoids â&#x20AC;Ś etc.

They were also familiar with volumes of some solids such as: Cubes, parallelepipeds, truncated pyramid.

pyramids,

and

An interesting thing: is a tablet gave the wrong volume formula for truncated pyramid. However, another tablet provided the correct formula: Volume = (A1 + A2 + A1 â&#x20AC;˘ A2 )(h 3) or its equivalent, where A1 and A2 are the areas of the two bases of the truncated pyramid and h is its height.

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Figure 08.2.

Truncated pyramid

The Mesopotamians were also familiar with geometric facts such as: The line joining the midpoint of an isosceles triangle is perpendicular to the base. The corresponding sides of similar triangles are proportional. The theorem today as â&#x20AC;&#x153;Pythagorean theorem,â&#x20AC;? The angle subtending a semicircle is a right angle.

Every culture has some fixations of ideas or mania of fashions. These conventions exist in literature, science, as well as in mathematics. Mesopotamian mathematicians were not immune of fixations and fashions. One of those things was that they did not seem to care for triangles that are neither isosceles nor right angle triangle. A non-

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isosceles, non-right angle triangle was never mentioned in any tablets as far as archeologists know. Another Mesopotamian fashion is that they consistently drew isosceles triangles, “standing” with their bases “vertical,” not “sitting” on them as we usually do. See figure 08.2. Sure, it does not make a difference; our society has a fashion of depicting things and Mesopotamians had theirs.

Figure 08.2.

Isosceles Triangle

Incidentally, this fashion of a culture may partially explain the reason for the Mesopotamians did not write down the methods by which they obtained their results. It might have been the fashion of their times, while our societies require us to publish even the rubbish otherwise we perish! We will

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elaborate on this and related topics in

CHAPTER

10.

Back to actual geometry! The Mesopotamians gave the area of a circle to be 3/4 of the area of the circumscribed square. Thus, they took the number denoted nowadays by π, to be three. It is worthwhile to note that at about 500 BC the Bible30 gave π also to be three. Did the Mesopotamians influence this claim? No comment. However, a more recent excavated tablet dated at about 2000 BC, claimed that:

The perimeter of a regular hexagon The circumference of circumscribed circle = .[57][36]. ! !

Since, the number .[57][36] (in the sexagesimal system) is the same as .96 (in decimal system), therefore, according to this claim: 6 /(2 π) = 0.96. Hence, π = 3/.96 = 3.125, (in decimal system), which is not bad! It is correct to one decimal place.

See:

1 Kings 7:13

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Figure 08.3.

A circle with an inscribed hexagon

The Ancient Egyptians at about 2000 BC had π as 3.1625. This is again correct to one decimal place.

08.4. KNOWLEDGE OF PYTHAGOREAN THEOREM

The theorem that today we call “Pythagorean theorem” was well known in both Mesopotamia and Ancient Egypt, at least as an empirical fact. As an example, there is a Mesopotamian tablet on which inscribed the following problem and its solution: 4 is the width and 5 is the diagonal, what is the breadth?

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The solution was given as follows: 4 times 4 is 16. 5 times 5 is 25. You take 16 from 25 and there remain 9. Now what times what is shall I take to get 9? 3 times 3, is 9. 3, is the breadth.

Both the Mesopotamian and Egyptian surveyors utilized the triple (3, 4, 5) to erect perpendiculars on the ground and in space. In fact, the surveyors of Ancient Egypt were described as “rope-stretchers.” They sold their skills by stretching their ropes. One also notices the “eternal” fact how a mathematical discovery is a source of income for engineers

Figure 08.3. The Rope-Stretchers. Observe 3,4 and 5 knots on the rope

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No wonder Mesopotamian mathematicians would look for other Pythagorean triples. This will be considered in the next chapter.

08.5. YALE TABLET In CHAPTER 06, there was a passing mention of Tablet Yale YBC 7289. This is another important mathematical tablet. It looks like a school problem. At the price of repetition, the figure is shown again on the next page. A square with both diagonals drawn on one side of the square is written the number [30]. Along one of the diagonals is written [1][24][31][10] and below it [42][25][35] is written in cuneiform with no sexagesimal point.

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Yale Tablet YBC 7289

Figure 08.4. = Figure 06.5

Now

multiplying

[30]

[1].[24][51][10], one obtains the second number that is recorded on the tablet, namely

[42].[25][35], which is nothing but the diagonal of the square shown on the tablet. This demonstrates that the Mesopotamians had an excellent understanding of Pythagoras Theorem, and that they had good method for approximation of 2 . Refreshing the readerâ&#x20AC;&#x2122;s memory, we had made a lengthy calculation in CHAPTER 06 using the approximation formula: 1+ a =app 1+ a 2 obtaining the same number that appears on the tablet namely [1].[24][51][10] as an approximation for 2 .

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We had found that the approximation was correct to three sexagesimal places.

08.8. SUSA TABLET Susa is a town in present day Iran. However, in the ancient times it was a Mesopotamian city. Susaâ&#x20AC;&#x2122;s tablet poses the problem of finding the radius of a circle circumscribing an isosceles triangle of lengths [50], [50], and [1][0]. The tablet registered the correct answer [31].[15]. Therefore, it is clear that the Mesopotamian had a good mastery of geometry. The tablet shows that they, besides being familiar with the Pythagorean theorem, knew that the line joining the vertex of triangle to the midpoint of the base is perpendicular to the base and passes through the circumscribing circle. The following figure provides a partial illustration of the problem.

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[50]

Figure 08.5. The “gray small” spot represents the center of the circumscribing circle. The circle itself is not shown in the figure. The length of the base is [1][0]. Let us carry the calculation utilizing the sexagesimal system. Denote the required radius by R. The length of the perpendicular falling the base of the “large” triangle = [50]2 " ([1][0] [2])2 . Use the table of squares.

= [50]2 "[30]2 = [41][40] "[15][0]

= [26][40] = [40] . Consider the “black” triangle. ! The length of its hypotenuse = RThe length of its base = [1][0] [2] = [30]

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The length of its other side

= [40] – R .

Appealing to the “Pythagorean theorem”, we obtain: [30] 2 + ( [40] –!R) 2 = R2 [30] 2 + [40] 2 – ( [2])( [40] ) R + R2 = R2 ( [2])( [40] ) R = [30] 2 + [40] 2 Hence, R = ([30]2 +[40]2 ) ([2])([40]) ! ! = [31].[15].

! !

194

The reader is requested to perform the calculations in both the decimal and the sexagesimal systems of

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09.

Geometric

Arithmetic Sherlock Holmes in Babylon â&#x20AC;&#x201C; R. C. Buck

09.1. INTRODUCTION The baked clay tablet with the catalogue number 322 in the G. A. Plimpton Collection housed at Colombia University may be the most known mathematical Mesopotamian tablet. It is dated at 1800 BC plus or minus 40 years. It shows the most advanced pre-Greek mathematics and contains a classification of Pythagorean triple. Thus, it is the most scientifically significant

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tablet (discovered so far) and the most photographed one. Neugeauber and Sachs were able to decipher it back in 1945. See [N] and [NS]. The tablet is slightly broken, where it is unclear what all the entries are.

Figure 09.1 Picture of Plimpton 322 Tablet

09.2. WHAT IS A PYTHAGOREAN TRIPLE? A triple (a, b, c) of positive integers is called a Pythagorean triple if a2 + b2 = c2. This means that there is a right angle triangle, with a hypotenuse of length c, the other two sides of lengths a, and, b, where a, b, and c are positive integers.

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c Angle ! b b

Figure 09.2.

A Pythagorean triple if each of a, b and c is an integer

The most familiar example of Pythagorean is the triple (3, 4, 5) since 32 + 4 2 2 = 5 . The triples (5, 12, 13) and (65, 72, 97) are other examples, but the triple (2, 7, 11) is not since 22 + 72 does not equal 112.

The famous (or may be the Pythagorean-Brotherhood-infamous) triple (1, 1, 2 ), which, in spite of the fact that 12 + 12 = ( 2 ) 2, is not a Pythagorean triple because 2 is not an integer (in fact it is an irrational number.) It is obvious that

! !

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Let d is a positive integer. The triple (a, b, c) is Pythagorean If and only if the triple (d x a, d x b, d x c) is Pythagorean.

A Pythagorean triple (a, b, c) is described as primitive if the greatest common divisor of a, b, c is one. That is there is no common factor among the three numbers a, b and c except one. The triples (3, 4, 5), (5, 12, 13) and (696, 697, 985) are primitive Pythagorean triples. The triple (45, 60, 75) is not primitive since the number 15 is a common factor among the three numbers.

09.3. IMPORTANCE OF PYTHAGOREAN TRIPLES We have seen in the previous chapter that the Mesopotamians (as well as the Ancient Egyptians) were familiar with the special Pythagorean triple (3, 4, 5). Their surveyors had utilized its properties in

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constructing perpendiculars on lines lying on the ground as well as in space. This particular triple is primitive. Then a question arises: Are there triples?

primitive

Pythagorean

The answer, of course, is yes. We have already given three examples of such pairs. A sharper question would be Is there a scheme for generating all primitive Pythagorean triples?

The answer is again yes. In general, theorems that provide methods for generating some mathematical phenomenon are described as classification theorems.31 Thus, the positive answer for the above question constitutes an important classification theorem. The proof of the classification theorem of primitive Pythagorean triple is usually 31

Classification is important not only in mathematics, but also in everyday life. For example, classifying shirts (in a clothing store) according to color and size is superior and more efficient, than leaving the collection of in the store shirts in a huge heap.

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credited to Diophantanus, (about 200 AD â&#x20AC;&#x201C; 284 AD), a Greek mathematician who lived in Egyptâ&#x20AC;&#x2122;s Alexandria and contrary to the Greek tradition at that time, did some works in algebra and arithmetic. The importance of the baked clay tablet Plimpton 322 lies in the fact that it contains a table of 15 primitive Pythagorean triples. Though the table is too short, yet it indicates that its author knew the classification theorem and most likely had a proof. It points out he or she had a deep understanding of mathematics and its ramifications. The tablet shows that the unknown Mesopotamian author had preceded Diaphanous by about two thousand years! Therefore, the tablet discloses the most advanced pre-Greek mathematics and demonstrates the level of sophisticated mathematical knowledge enjoyed by Mesopotamian mathematicians. The statement of the classification theorem will be provided in the next section. Meanwhile we feel that some of the readers are restless and are silently demanding a response to a query in the back of their minds which say: â&#x20AC;?Well, we are now convinced that the tablet contains a good piece of mathematics. However, for those ancient

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people of Mesopotamia who are supposed to be pragmatic people, is not one or two Pythagoras triples enough for their practical purposes? Whatâ&#x20AC;&#x2122;s the importance of searching for other such triples?â&#x20AC;? We are not sure what the true answer for this question is. There is nothing in tablet Plimpton 322, or in other excavated tablets, which are known so far, which indicates the reasons for listing several Pythagorean triples. We can only put forth conjectures. One observes that to each Pythagorean triple there corresponds an angle Ď&#x2022; as shown in Figure 09.2. There corresponds for the particular Pythagorean triple (3, 4, 5) an angle about 36.87 degrees. It is conceivable that for some structural-engineering purposes the particular angle of degree 36.87 was inconvenient. There might have been some kind of an obstacle that made the triple (3, 4, 5) or its multiples impractical. Other essentially different triples were badly needed. Thus, there was an open unsolved mathematical problem with roots in practical reality. The problem might have remained unsolved for a couple of decades or centuries, until a mathematically minded Mesopotamian set out to investigate that dilemma. Because of his or her study a method, which must be envied by modern mathematicians, was

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developed. The outcome did not only provide an answer to that particular engineering need but also produced by one stroke, a general method for generating ALL Pythagorean triples. Lo! We have Plimpton 322. Of course, this is all conjecturing. Nevertheless, in reality, Mathematics evolves that way. A mathematically inclined mind plunges into the deep sea of the unknown and often brings back beautiful pearls. On the other hand, a mind not mathematically inclined prefers to stay near the shores of the unknown.

09.4. HOW TO GENERATE PYTHAGOREAN TRIPLES?

PRIMITIVE

In the section, we will present the method that assures the existence of a method for generating primitive Pythagorean triples. We will only state the theorem. The complete proof of the theorem, though elementary (in the sense that it does not need mathematical “artilleries” beyond high school mathematics) is lengthy. Therefore, it is preferable that it be “exiled” into APPENDIX (MATHEMATICS).

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Theorem 1. Let (a, b, c) be a Pythagorean triple. If the triple is primitive then c has to be odd and either a, or, b is odd. 2. A triple (a, b, c) is a primitive Pythagorean triple where a is odd and b is even if and only if there are positive integers p and q that have no common factor except 1 such that a = p2– q2, b = 2pq, c= p2 + q2

By appealing to this theorem, the reader can “cook” as many primitive Pythagorean triples as wanted. Just pick any two positive integers, one of which is even, the other is odd and having no common factor except one. This is not a difficult task. Calculate a, b, and c as the theorem indicates. Then the triple obtained is primitive Pythagorean. Let us do some sample “cooking” ourselves: Take p = 2 and q = 1. Then the triple (22-12, 2 X 2 X 1, 22 +12) is our “old great friend”, namely (3, 4, 5). • Take p = 4 and q = 1. Then the triple (42 -12, 2 X 4 X1, 42 +12) = (15, 8, 17) is a primitive Pythagorean triple. • Take p = 125 and q = 64. Then the triple (1252 – 642, 2X125 X64, 125 2 + 642 ) = (9529, 16000,19721). •

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This particular triple will be needed in the discussion about Table 09.1.

The table, on the next page, is a reconstruction of the Plimpton 322 tablet with the numbers expressed in the decimal systems. The first column on the right gives the number list for the fifteen rows. The second and third columns on the right contain the entries of the parameters p and q. The third, fourth and fifth column contain the entries of the three numbers a, b and c. the first column on the left is the angle of the right angle triangle opposite the side of length a.

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Listing of rows

120

4825

2456

4601

6649

4800

43.27

12709

18541

13500

125

42.08

41.54

319

481

360

40.32

1291

3541

2700

39.77

799

1249

960

38.72

481

769

600

37.44

4961

8161

6480

36.87

35.78

9529

19721

16000

34.98

1679

2925

2400

degrees

=p2â&#x20AC;&#x201C;q2

=p2+q2

= 2pq

44.76

119

169

44.25

3367

43.79

Angle

a b

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33.86

161

289

240

33.26

1771

2959

270

31.49

106

Table 09.1 Plimpton 322

‘Modernized’

The tablet demonstrates that the Mesopotamians must have had extensive trigonometric tables that enabled them to calculate angles in degrees correct at least up to two decimal places. Moreover, the numbers p and q are so cleverly chosen that their prime divisors of each are 2, 3 and 5 only. Remember that these three numbers are the only prime factors of sixty. 2 Namely, 60 = 2 x3x5. These choices might have been made by the author of the tablet, in order to, avoid confronting recurrent fractions that might appear in the division by b’s needed in the calculations of angles. The column of angles shows that the angles are spaced by approximately one degree from each other. The author of the tablet might have had a certain purpose in mind that made him or her choose p’s and q’s such that the set of the corresponding angles are almost uniformly spaced and ranges from about 30º to about 45º.

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E. C. Zeeman in his research paper [Z] considered all possible pairs p and q satisfying the conditions: (1) Required by the theorem. (2) The prime factors of each p and q are 2, 3 and 5. (3) The set of associated angles ranges between 30 and 45 degrees.

He found an amazing result, that there are exactly sixteen such pairs; fifteen of them were listed in the original tablet. However, there is one pair missing. The missing pair is included in table 09.1 in the unnumbered â&#x20AC;&#x153;dark grayâ&#x20AC;? row. The scriptwriter might have been drowsy and forgot to record the missing pair. We will have comments in the next chapter about the impact of this tablet on the controversial problem of whether or not the Mesopotamians possessed the notion of a proof. There must be some readers, who are interested in, detailed information about Plimpton 322 tablet and its ramifications. We refer those readers, to papers by the E. M. Bruins [Br], R. C. Buck [Bu] and E. C. Zeeman [Z].

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10. To Prove or Not to

Prove. That is the

Problem The proof of the pudding is the eating â&#x20AC;&#x201C; Proverb

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10.1. INTRODUCTION Shakespeare would have probably portrayed the problem that confronts us by rephrasing his famous saying. In fact, there is not only one problem, but also at least four: What are the criteria for an argument to be accepted as a poof of a statement? Did the Mesopotamian mathematicians possess the concept of proof? If they did, then why didn’t they record the proofs of their results? If they did not wish to “publish” their proofs, then is it possible to guess some of their proofs?

This chapter deals with these and related problems. It seems that many mathematicians consider a mathematical statement to have been proved if the arguments for its truth satisfy the Euclidean criteria of proof. Though we believe that the Greek achievements were real revolutions in human cognition and constituted a significant historical paradigm shift, but to put the criteria for acceptability of proof according to the

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Greek norms is too severe and too narrow. Let us first briefly describe the notion of proof in the Greek paradigm. 10.2. THE GREEK NOTION OF PROOF •

•

• •

210

Few terms were taken without definition. They were called undefined notions. Few initial propositions about these terms were taken as true; called axioms or postulates. The undefined terms and axioms were the cornerstones upon which the axiomatic method was built in the following way: New terms were defined by using the undefined notions. A body of statements is generated by both the undefined and the newly defined terms. These statements were arranged in a linear order in such a way that each can be derived from the previous statements or from the axioms by pure logic. These statements were called theorems.

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â&#x20AC;˘

Moreover, in the geometry of Euclid there was the requirement that all geometrical figures to be constructible by straight edge and compass only.

Thus, we see for the first time in history a concern for consistency and a systematic response to that concern.

10.3. TOO MAGNIFICENT STRUCTURES FOR THE REAL WORLD

Yes, the logical structure was so magnificent that the Greek mathematicians had fallen passionately in love with it. Without being aware, they became slaves of their own methods. They imprisoned themselves inside the logical edifice they had built. The glaring beauty of the logical structure that they had built blinded even the Greek intellectual giants, such as Archimedes 32. They did not stumble on the

Thanks to Ms Barbara Hubbard, who brought up the point that Archimedes calculated Ď&#x20AC; to a great accuracy. This is true, but he made his

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concept of zero, and they could not see the advantages of the place-value system, which was right under their noses, used in neighboring Egypt, Mesopotamia and India.33 The Greeks showed concern for the logical structure of mathematics. The Pythagorean school as was explained in CHAPTER 05, sought to put the foundation of all of mathematics on the set of rational numbers but were bewildered by the discovery of incommensurable ratios in geometry. This discovery prevented the Greeks from giving an account of geometric magnitudes in terms of only rational numbers. Instead of reexamining the foundation of their philosophy, they arrogantly raised their heads up and never intended to look at the source of difficulty again. Thus, they almost discarded the study of algebra and arithmetic. Moreover, “Plato’s ideals” made mathematics as a subject of almost out of this world.

calculations by using the Greek numeration system and by the place-value system. 33

The Greek astronomer Ptolemy used the Mesopotamian sexagesimal system.

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10.4. EURO-CENTRIC VIEW HISTORY OF MATHEMATICS

THE

The commonly held view among many mathematicians in the west is that the discipline of mathematics was founded during the period 600 BC â&#x20AC;&#x201C; 300 BC. Consequently, they tend to ignore, trivialize, or minimize the achievements of the older civilization of Mesopotamia, Egypt, India, and China. For those Western mathematicians no real mathematics existed until 600 BC when Thales of Miletus initiated the notion of proof. Consequently, Western mathematicians think they owe nothing except to the Greeks. So much so that they concede only to the part of Arab-Muslim scholars as custodians for the Greek legacy ignoring the significant contributions of the former scholars. They also tend to ignore that Thales himself, as well as his student Pythagoras were disciples of the priests of both Egypt and Mesopotamia. It is true that the Greek achievements transformed the discrete collection of knowledge into a well-knit coherent structure. This is an achievement, whose immensity no person, in his or her right frame of mind, can deny. However, its greatness should not nullify or trivialize the role of other cultures.

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It should also be remembered that other cultures possessed different norms from those of the Greeks. In sections from 10.5 to 10.9, we will refresh our memories of a few outstanding Mesopotamian contributions that were dealt with in the previous chapters. We will try to show that behind those contributions were first-rate outstanding arguments, proofs, corroborations, or whatever one calls them. In Sections10.10 and 10.11, we will consider the problem of why the Mesopotamians and other ancient civilizations did not publish their proofs. Section 10.12 is a concluding section in this chapter.

10.5. EXAMPLE: PRIMITIVEPYTHAGOREAN-TRIPLES THEOREM Some impressive theorems that were known for thousands of years before the era of the Greeks were credited to Greek mathematicians. We cited as an example the classification theorem about the primitive Pythagorean theorem depicted in Plimpton 322 tablet. It is usually credited to the Greek Diophantus. However, an unknown and unnamed Mesopotamian mathematician had

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preceded Diophantus by about two thousands years! The unknown Mesopotamian author seems, in today’s perspective, to be well versed in mathematics and possessing a mental agility that enabled him or her to match the theoretical results obtained, with the practical problem at hand. This matching appears to necessitate that the associated angles to be between 30º and 45º. Furthermore, the author was evidently aware that there was an infinite set of solutions. This infinitude enabled the author to choose parameters p and q to avoid the complicated division. Briefly, the Mesopotamian author knew what he or she was doing. Therefore, it is inconceivable that the author of Plimpton 322 tablet had obtained the results through empirical means. The proof of the classification theorem, provided in APPENDIX (Mathematics), though lengthy, is not difficult, and within the capabilities of the Mesopotamian mathematicians. When we say that the proof is not difficult we mean that it does not need mathematical “weaponries” beyond high school mathematics!

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10.6. EXAMPLE: A PROOF OF THE PYTHAGOREAN THEOREM Another example is the famous Pythagorean theorem. This theorem was well known in each of the four main ancient civilizations of (Mesopotamia, Egypt, India and China) and had been extensively utilized in their constructional works. It is true that no proof was recorded. Nevertheless, this does not mean that no proof existed or no proof was known at that time. Consider for example Figure 10.1. Figure 10.1

The pattern in the above figure appears on a Mesopotamian tablet, housed at present, in the British museum. A mathematical â&#x20AC;&#x153;Sherlock Holmesâ&#x20AC;? may find in that tablet a clue for a proof of Pythagorean theorem! The

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beautiful pattern, with colors, whose picture is shown in Figure 10.2 on the next page, reflected the ideas in the clay tablet. The figure drawn on the clay tablet might have been copied from patterns on the tiles covering of the floors of many Mesopotamian temples. Anyway, one can observe in the figure that there are an inner and an outer square. The vertices of the inner square lie on the sides of the outer square. Thus, four congruent right angle triangles surrounding the inner square are constructed. Hence, each of the four sides of the inner square is a hypotenuse of one of the four right angle triangles. The pattern would certainly impress any ancient (or modern) mathematically minded person that it is “pregnant” with a remarkable geometric identity. The following is a fictitious story of how the geometric identity was delivered without any “difficult midwifery.” After a couple of thousands of years that newborn identity was hijacked and was named by baptism as “Pythagorean Theorem.” Let us proceed with our story: The pattern shown in the figure below decorated the whole floor of a temple situated high on the top of a ziggurat. Accordingly, that configuration was in the front

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of the eyes of the Mesopotamian priests and priestesses while they were collectively praying, meditating at the temple and chanting the classical monotonic hymn manta â&#x20AC;&#x153;Ya Aleel, Ya Anuâ&#x20AC;? imploring Aleel and Anu, the gods of the temple.

a Figure 10.2.

218

A Pattern that suggests a proof of Pythagorean theorem

- 219 -

Among those worshippers was a certain mathematically minded priest. His name was Thoro. His colleague priests envied him because he had an ability, which they did not have. He was known to be able to do calculations in his head; he knew the 3431 entries of the Mesopotamian-sexagesimalmultiplication table by heart. Thoro – as the rest of the priests – saw that pattern every three hours, day in and day out, month in and month out, and year in and year out. Yet, it did not seem to have any tangible influence on his conscious mind. However, he, on one early morning, after having a sleepless night, was as usual meditating and chanting with the rest of the priests: “Ya Aleel, Ya Anu”. Suddenly the pattern seemed to punch hard on his unconscious mathematical mind. The pattern set his psyche working. His lips were reciting the mantra, but his brain was working on a different plane, mentally performing calculations related to the pattern. He began to speak silently to himself, “Denote the lengths of the sides of the right angles of the four triangles by a and b, and the lengths of their hypotenuses by c. Hence, the length of each of the

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sides of the inner square is c, and each of the sides of the outer square has length a + b.” He silently continued: “Well, the area of the outer square 2 is (a + b) . It also equals the area of the interior square plus the area of the four right triangles, Therefore (a + b) 2 equals 2 c + 4(ab/2). From which he concluded that a2 + 2 ab + b2 equals c2 + 4 ab /2. He was seeing the equality glowingly highlighted in front of his inner eyes. He continued his silent soliloquy, “Simplifying leads to a2+b2 = c2 ” “That is a great newborn result!” He must have murmured the last sentence in an audible voice because; it was observed that the two priests standing next to him turned their heads toward him for few seconds. As the walls of the temple echoed the sounds of ringing bells, other bells were ringing in the mind of Thoro. His lips were still reciting the mantra. His mind was elsewhere making a dialog with itself. “Had I not seen surveyors and engineers erecting right

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angles by dragging their (3, 4, 5) knotted ropes around. Yes! Those robes are nothing but special empirical cases of my new result. Oh, yes it is. This means that. In any right angle triangle, the sum of the squares of the lengths of the sides a right angle equals the square of the length of its hypotenuse.” After that, there were periodical lapses of a few minutes, during which he was able to regain control over his mind and join the rest of the group in their chanting of “Ya Aleel, Ya Anu.” During a lapse in which he lost management over his conscious mind, he questioned himself in a silent soliloquy: “Wait a minute! Don’t be hasty.” He commanded himself and continued, “Unless, I can build from any given right angle triangle a pattern like the one the floor then my statement is true for all right angle triangles.” He stopped for a minute, and then silently asked himself, “Can I do that?” After a few lapses, he soundlessly answered the question. “Yes. Certainly I can, it is so easy.” He continued, ”Given a right angle triangle, let me

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construct four exact replica of the triangle, and arrange them in a cyclic order such that each intersects only two of them in exactly one vertex and in such a way that one is a mirror image of the other two.” At this moment, he forgot all about the gods of the temple. He ran down all the hundred and fifty steps from the top of the ziggurat to the nearest scribe. Then he took a lump of wet clay and drew the diagram shown in figure 10.1, as a reminder or as a patent (as they are called today.) He turned back and slowly climbed the one hundred and fifty steps up to the top of the ziggurat, only to find his priest colleagues finishing their worship. The other priests thought that his holiness Thoro was getting crazy. He must be, otherwise how could he be able to do all that calculation in his head while praying. Consequently, he was excommunicated.

The fictitious story ends. The story – though being a history fiction – reflects something analogous must have occurred. The story was meant to drive the ideas in an entertaining and interesting manner.

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Do not the arguments presented by our hero-priest constitute a proof of Pythagorean theorem? We believe it is. The disciples of Euclid might reject the proof presented, What Euclid did in his Elements was to banish ALMOST anything algebraic out. He put the known results of that time in a linear order so that any statement (except the axioms and postulates) can be derived from the preceding statement or the axioms. Greek mathematicians, being prisoners of their own methods, would probably reject the above arguments articulated by the priest as a proof, since it uses algebraic tools. The Mesopotamians, Egyptians Indians, Chinese, and the Arabs did not have such inhibitions towards algebra or arithmetic. All tributaries to mathematics were acceptable to them. They would accept the argument as sound proof.

10.7. EXAMPLE: METHODS

exact

APPROXIMATION

Although mathematics is mainly an science, but there are situations

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(especially when dealing with problems that arise from real life) in which approximations are needed. In such cases, one has to preassign an accepted tolerance of error for the problem at hand. Mesopotamian mathematicians were skillful in their use of approximations. They had constructed many tables for different purposes. In their construction of tables, they likely employed the formulae or their equivalents that were mentioned in CHAPTER 06. The number [1].[24][51][10] that appeared on Tablet YBC7289 was nothing but an approximation of [2] correct to three sexagesimal places. They must have obtained it by successive approximations. The accuracy of their result was demonstrated in Section 06.7 through ! successive approximations by using the formulae cited in CHAPTER 06 Mesopotamian mathematicians were fond of exploiting inverses in their performing the operation of division. The approximate formulae provided in 2 CHAPTER 06 presuppose that that a is less than the pre-assigned margin of error. In order to refresh memories we repeat the proofs given in that chapter: 2 Since (1 + a) (1 – a) = 1 – a , therefore, 2 by disregarding the a term we obtain app 1 / (1 + a) = 1 – a.

224

- 225 2

Since (1 + a / 2) = 1 + a + (a / 2) , 2 therefore by disregarding the a term we obtain 1+ a = app 1+ a 2 . Similarly for obtaining 3 1+ a = app 1+ a 3. We believe that these proofs were likely to! have been known to Mesopotamian mathematicians for otherwise how could they !obtain the! good approximations they had obtained. Proofs are NOT restricted to geometry!

10.8. EXAMPLE: SOLVING EQUATIONS Consider another example. Mesopotamians mathematicians solved the quadratic equation x2 + bx = c, where b and c are positive numbers. They solved the quadratic by using the method of completion of the square algebraically. The ingenuity of those ancient mathematicians is shown through possessing alternative methods for attacking their problems. We have seen several examples in Chapter solving quadratic equations and simultaneous equations. A possible alternative method is a geometric completion of the square, the process of which can also be depicted via the diagrams shown in Figure 10.2

225

- 226 -

from which the solution x = "b 2 + c + (b 2)2 is obtainable. Does not the reader agree that this method is ingenious whether accepted by Mr. Euclid or not? ! Take the example of the brave attempt of Mesopotamian mathematicians at the solution of cubic equations treated in section 07.2. They did not construct the table of 2 3 n + n just for fun. There does not seem to be any purpose other than dealing with the cubic. Moreover, a great ingenuity is shown in the process of reducing a cubic to 2 3 the form n + n . It was a brave step towards solving their problems. Keeping in mind that the algebraic solution of the cubic equation was completely accomplished as late as the 16th century AD, this makes it imperative that the courageous Mesopotamian steps should be appreciated and credited.

226

- 227 -

x X mmmm

b Area =bx

A1 Area A1 = x2 + bx =c

b/2

x A2 Area A1 = Area A2 =c

b/2

x + b/2

Area A3 = ( x+(b/2) )2 = c + (b/2)2 b/2

Area = ( b/2)2

Therefore

x + (b/2) = !c+(b/2)2

Figure 10.2. Completion of the Square Method

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10.9. PIE(S)

EXAMPLE:

SEQUENCES

AND

The same statement could be made about the sums of the finite sequences dealt with in SECTION 07.8. The mere leaving the right hand sides of the two sequences as it is 2 9 9 9 1 + 2 + 2 + … + 2 = 2 + 2 –1, 2 2 2 2 1 +2 +3 + … +10 = (1 x (1/3) + 10 x (2/3)) x55, with no further simplifications, indicate that the mathematicians of Mesopotamia had a scheme for reaching out for the summation. In other words, they likely had a proof. It is true 3 was not a good estimate for the number π. Yet, a mathematical Sherlock Holmes notices that the estimate was given as a ratio of the perimeter of a hexagon. This is an indication there was of a kind of planning for an attack. The improved estimate for π, as: Perimeter of the 12-sided polygon / (2π) = [0].[59][36], which amounts to getting π = 3.125, was attempted again as a calculation of the perimeter of a twelve-sided regular polygon. In fact [0].[59][15] approximates cosine of 15 degrees. The computation

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required the finding of the trigonometric functions of half angles. The method of approximating a circle by regular polygons carried to a great degree of accuracy by Archimedes, Anyway, the method of the Mesopotamian mathematicians was a brave initial assault on the unknown.

We gather from the preceding discussions that: It seems that the Mesopotamian did not lack the notion of proof. They lacked a mathematicians-librarian (like the eminent Euclid) who would have extended his main occupation of arranging manuscripts into arranging the known mathematical proofs and facts in a linear order. To arrange the known mathematical facts he has, in order to start with, to have some undefined terms and some initial statements. If the Mesopotamians have had such a librarian-mathematician, they would have hit, a thousand years earlier, on the goldmine of axiomatic system! This leads us to the dilemma of the lack of publication of proofs. 10.10. WHY DIDN’T THEY ‘PUBLISH’ THEIR PROOFS?

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Again, as usual, the answer is not known for sure. However, we get some illumination, if we have a critical look on our today’s modern society! The general attitude towards disregard to learn about proofs in today’s computeroriented society provides some indication for a possible partial answer to the above query. It is a fact that most modern people, with the possible exception of university professors of mathematics, DO NOT care for the actual proofs of formulae obtained for professional practical purposes. Moreover, many of today’s scientists DO NOT ask about the programs and the algorithms behind software such as “Mathematica” or “Maple” that deal with methods for solving sophisticated problems of integration, differential and integral equations, and deals with complicated graphics. Even more many modern students had lost many childhood curiosities. They do not think or bother to inquire about the programs and algorithms upon which that software are built. Some of the university students would appeal to the famous saying that: “The proof of the pudding is in the eating.” when obstinately refusing to learn about these topics. They often argue that it is enough to

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use software as long as these are successful in supplying answers. Now, who knows? Mesopotamian consumers of mathematics might have had the very same attitude as modern students and engineers!

10.11. A FICTITIOUS STORY

Now let the reader free the reins of his or her imagination in order to meet the hypothetical Lumerian people of the year 4005 AD who inhabit a hypothetical country called Lumeria!

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Lumerian archeologists are greatly excited because they recently have excavated few compact discs, which they were able to date back t o two thousands years ago – precisely to 2005 plus or minus 10 years. These few compact discs are preserved as precious treasures in the museums or libraries of some large centers of Lumeria. The most eminent scientists are cu rrently studying and s crutinizing those discs. Some grea t thinkers of Lumeria were able to decipher the discs. They find out that the discs contain many deep mathematical procedures among w hich are methods for solving complicated differential equations and i ntegral equations, and some results apparently built on non-Cantorian set theory. The Lumerian scientists were shocked to discover that a couple of the results found on the newly discovered discs were only recently proved in the years 3985 and 3994 and the proofs were already attributed to famous Lumerian mathematicians who were awarded the Green Domain medal, the highest honor given to mathematicians in fourth millennium era. Naturally the great scholars of 4005 started to wonder about the deep results deciphered in the excavated discs: How did those “primitive people” of 2005 obtain their results? Were the people of 2005 able really able to prove those wonderful results? Or they just stumbled upon them by me thods that are not acceptable to the fourth millennium era? If the people of 2005 had really proved those results then why, on earth, didn’t they “publish” their proofs? Few open-minded Lumerian thinkers conjectured that the majority of software consumers of the era 2005 might have been so pragmatic and so practical that they rarely demanded proofs. So much so that the mathematicians of 2005 published the proofs and the algorithms only on perishable papers and not on compact discs. May be one day said the open-minded Lumerian thinkers will stumble on some of those perishable papers which might shed on the mystery. “Amen”! Was the response. Those few open-minded Lumerian thinkers would point out to their colleagues that the mathematicians of 2005 were writing things and communicating with their own people of 2005 in their minds; it had never occurred to them that future people of two thousand years later will one day be peeping at their results! “Therefore” the open-minded Lumerian thinkers add “one should NOT impose the standards or norms of publications of the fourth millennium era on the standards of the ancient civilizations of the second millennium era! Let us recognize the cultural context of the problems 232 tackled, with eyes open looking for similarities that are shared.”

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10.12. CONCLUSION Is not this fictitious story analogous to the situational relationship between Mesopotamian achievements and our modern society? The quality of mathematical production of a civilization cannot and should not be measured by comparing it with the mathematics of another civilization. The two may not have necessarily a common measure. They may be incommensurable. One has to recognize the cultural context of the problems tackled, with eyes open looking for similarities that are shared. With this outlook, mathematics of all the ancient civilizations (Mesopotamian, Egyptians, Indian, Chinese, Mayan and others) is seen not only as predecessors for

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the Greek mathematics, but also as predecessors for our modern era too. We should see all those civilizations as our associates, and colleagues in scientific endeavors. With such a viewpoint, Mesopotamian mathematics, even if it were mere collections of practical rules and useful techniques, lacking the notion of proof, would present itself as a remarkable pursuit. Thus, Mesopotamian mathematics is an activity â&#x20AC;&#x201C; alongside with cuneiform writings, legal laws of administration, mythology, religion, astronomy and literature â&#x20AC;&#x201C; that played roles (mentioned in CHAPTER 02) for unifying the many ethnic groups of people inhabited, immigrated or even invaded the land of Mesopotamia. With this approach in mind, the talk about existence or lack of existence of proofs (in the Greek sense or in any other sense) for the theorems and procedures used in Mesopotamian mathematics becomes irrelevant.

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11.

Astronomy The fault, dear Brutus is in ourselves, not in our stars â&#x20AC;&#x201C;Shakespeare

11.1. INTRODUCTION Mesopotamians astronomers might not have agreed with the above Shakespearian quotation. In fact, they had lived under the timeless, cyclic canopy of the starry sky and kept watching the stars for centuries recording their movements, partly to foretell the future. From which sprang the belief of the pseudoscience of astrology, which survived even in todayâ&#x20AC;&#x2122;s society. Mesopotamians made astronomy, the first of the sciences by devoting a period of about a thousand year in its development. It is

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unequaled in modern “Western Science.” Some historians date science back to Galileo, Kepler, and Newton. This would be less than 500 years. In fact, Swerdlow describes in his book “The Babylonian Theory of the Planets ”[S]: “…. [Mesopotamian] Astronomical Diaries, originally extending from the eighth or seventh to the first century BC are by far the longest continuous scientific records or should we say the record of the longest of any kind in all history, for modern science has existed half as long”

Keeping records of celestial phenomena was quite important for understanding of events in Mesopotamian society. Thus, by the seventh century BC, the professional astronomer scribes, had begun to register records of systematic observation of the sky without telescopes or timerecording instruments. These records are known today as ”The Astronomical Diaries.” Despite the extensive written records of Mesopotamian astronomy, there is very little knowledge about the instruments they had used and even less about the observatories, which must have existed. Clay “astrolabe,” housed in the British Museum in London is the only known

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available instrument. An astrolabe is an instrument used to measure the angular height of a celestial object. There are few references on tablets to some other instruments, like the water clock and the shadow stick.

11.2. TWO FAMOUS MESOPOTAMIAN ASTRONOMERS The Greek geographer Strabo of Amasia (64 BC – 23 AD) gave a description of Mesopotamian astronomy and mathematical astronomers. He named two prominent astronomers: Nabu-rimannu (? – 490 BC) and Kidinnu (? – 375 BC). Since there were abundant recorded observational data, the Mesopotamian astronomers were able to calculate accurately some astronomical constants. For example, the Nabu-rimannu calculated the length of the synodic month (that is, the period between two full moons) to be 29.530641 days. Later Kidinnu arrived at 29.530589 days, which is only 0.432 seconds more than the modern estimate. Kidinnu also made a calculation of the solar year with an error of only 4.5 minutes. Kidinnu calculated that the ratio of synodic month to the period between two

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consecutive moments when the moon is closest to the earth. He found the ratio to be 251/ 269. The calculation demonstrates Kidinnuâ&#x20AC;&#x2122;s observational skills. To honor Kidinnu the great Mesopotamian astronomer, a lunar crater of 56 kilometer in diameter was named after him The sources of all that astronomical information come from several Greek authors and astronomers. However, a tablet dated at about 370 BC, recorded that Alexander the Great beheaded Kidinnu during the period of his occupation of Mesopotamia.

11.3. WHAT WAS RECORDED IN THOSE DIARIES? The diary usually begins with a statement of the length of the previous month, which might have been 29 or 30 days. Then it records: The time between sunset and moonset on the day of the first waxing crescent. Similar information on the times between moonsets and sunrises and between sunrises and sunsets at full moon. At the end of the month, the interval between the rising of the

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last waning crescent moon and sunrise. The locations of the moon and planets with respect to the set of some thirty stars that the Mesopotamian astronomers used as reference for celestial positions.

When a lunar or solar eclipse took place, the diaries tabulate: The date of its occurrence. Its duration. The planets that were visible, at the time of its occurrence. The star that was culminating. The prevailing winds at the time of its occurrence. Significant points in the various planetary cycles. Reports of bad weather or unusual phenomena-like rainbows and halos are recorded. Various events of local importance, such as fires, thefts and conquests, prices of commodities, the amount of the rise or fall in the river.

11.4. MUTUAL RELATIONS: ASTRONOMY WITH MATHEMATICS

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It is no accident that the first civilization to advance astronomy was the first to advance mathematics. The Mesopotamian astronomers studied the successive sub-events of critical major astronomical events. Then they studied the sub-events back to the beginning of their records to learn how to predict them. They did this with the movements of the planets, as well as with the phases of the moon. Among the major events were the eclipses of the moon and the sun. These led to algebraic equations, which demanded solutions. These after being solved, the astronomer-mathematicians utilized the solutions in predicting future astronomical events. Their predictions were good except for the eclipses of the sun. This continuous feedback was of a great value to both astronomy and mathematics.

11.5. PREDICTIONS MOON DOES

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WHAT

THE

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These records were essential in the initial stages of the Mesopotamian astronomy. However, as the science of astronomy developed, the taking of new observations became less important. In other words, observational astronomy gave way to the mathematical analysis of the records of older observations, which in turn led to the mathematical predictions of current and future astronomical phenomena. At about the sixth century BC, the Mesopotamian astronomers, relying on the accumulated systematic data in their possession, were able to compute in advance the regularities of the motion of the moon for example: The expected period between moonrise and moonset and the expected periods between sunrise and sunsets for various days ahead. The position of the sun and the moon at new moon. The length of the daylight and the length of night. The rate of the daily motion of the moon through the stars and other related information.

The phases of the moon are quite important to a society, not lighted with electricity. A certain author puts it: â&#x20AC;&#x153;The nightsky was the only show in town.â&#x20AC;?

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The new moon allows a little more time in the day for gathering and provides more security from animal and human predators.

Planets received analogous attentions. However, their movements were not uniform as perceived from the earth. The Mesopotamian astronomers had to devise mathematical techniques that would take their variation of motion in account. In so doing, they made notable contributions to astronomy and had feedback improved the existing calendar. Moreover, they provided problems (as was mentioned earlier) to mathematicians to tackle. It should be mentioned, however, that the Mesopotamian astronomers did not develop the physics of the motion of stars and they did not suggest any geometric model of the universe. The astronomical problems were solved arithmetically.

11.6. PREDICTIONS OF ECLIPSES The extensive amount of available data in possession of the Mesopotamians astronomers enabled them to count backward

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and see how it got that way. They obtained some reliable results. A lunar eclipse was (and still is) frightening to many people. Thus, its prediction was helpful to the priests and scientists of the Mesopotamian society. Solar eclipses were even more frightening34! Using their data, the Mesopotamian astronomers were able to predict lunar eclipses (and later solar eclipses too) with good accuracy. The first recorded predicted solar eclipse was at June 15, 763 BC. The following is a picture of a tablet on which inscribed a list of eclipses.

Figure 11.1. Picture of a Tablet in the British Museum with a List of Eclipses Between 516 BC and 465 BC

There are recorded cases when a king, upon the prediction by his priests and astronomers of the occurrence of a solar eclipse, sent his soldiers to wait outside the enemy city and to take it over during the panic caused by the solar eclipse.

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11.6. CONSTRUCTION OF CALENDARS Astronomy evolved as a measurement tool. As the Mesopotamian society formed and became increasingly intricate, setting of calendars became vital for maintaining stability and development. One of the results of those astronomical records was nearly a perfect calendar. The Mesopotamian astronomers using those records were able to recognize that 235 lunar months are almost identical to 19 solar years. They concluded that seven of each nineteen years has to be leap years with an extra month. The difference is almost two hours. The names of months were: Mesopotamian name of the month

Arabic similar name of the month

Number of days

Mesopotamian name of the month

Arabic similar name of the month

Number of days

Nisanu Nisan

Aiyaru Aiy-yar

Arkhasmura

Kislimu

Dabtu

Simannu Duâ&#x20AC;&#x2122;uz Tamooz Abu

Ululu I Ay-lool

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30 29

Tashritu Teshreen 30

Sabadu Shibat Addaru I Adh-ar

30 29

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Ululu II Ay-lool

Addaru Ii Adh-ar

Table 11.1. The Names of Mesopotamian months and some similar names in Arabic.

The first day of Nisanu (that corresponds to our first of April) is the Mesopotamianâ&#x20AC;&#x2122;s New Year day, which is never far from the vernal equinox, which is the first day of spring.

11.7. MYTHOLOGIES BEHIND CELESTIAL OBSERVATIONS

THE

An inspection of almost all ancient societies reveals that their most important gods live somewhere in the sky. The regular motion of celestial bodies made the gods agents of order that helped giving meaning to the world down below. Their periodic appearance and disappearance suggested immortality. The light of celestial bodies drew attention and implied power. Moreover, being in the sky with such perspective on earth below, it was natural to assume that the gods must be omniscient because they could see every thing:

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Kenneth Spriggs! 10/4/05 9:55 PM Deleted:

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To see the world, oneâ&#x20AC;&#x2122;s eyes must be in heaven.

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Appendix

(Mathematics) SECTION 1. INTRODUCTION Appendix (Mathematics) contains a more detailed mathematical presentation than the other parts of the book. One of its purposes is convincing the reluctant reader that there exist other place-value numeration systems besides the familiar decimal system. Section 2 contains the statement of the theorem that constitutes the cornerstone of place-value system. Sections 3 and 4 illustrate the ideas presented through two examples: the binary and the duodecimal systems. Furthermore, the two sections provide examples of arithmetical manipulations within these two systems. Section 5 gives a complete proof of the classification theorem of primitivePythagorean-triples. This proof is included to be an indicator that the Mesopotamian

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mathematicians must have had some idea of the notion of proof. SECTION FOUNDATIONS SYSTEM

2.MATHEMATICAL PLACE-VALUE

The cornerstone of numeration is a theorem in mathematics that states: Let b be any positive integer greater than one. any non-negative integer z there is a non-negative integer k such that kFor (!) b ! z < b and k

k+1

(!!) z = ak bk + ak–1 bk–1 + ak–2 bk–2 + …+ a1 b1 + a0 b0, where each of the a’s is a non-negative integer less than b. When the number b is well understood by the society or understood from the context, all the b’s are usually suppressed as well as and all the plus signs from (!!). Then z is expressed simply as: ak ak–1ak–2 … a1a0.

We may add parenthetically that if a “bpoint” is introduced after a0, and if non-ending sequences of digits after the b-point are accepted then any non-negative real number35 35

Real numbers are, roughly speaking, the integers, the rational and irrational number. The adjectives, “rational” and ”real” describing the word number are misnomers that were inherited from our mathematical

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w can be represented as follows: w = ak ak-1ak…a1a0.a’1 a’2 a’3 …a’m–1 a’m … where each of the (a’)’s is a non-negative integer less than b . 2

For example, 15.789126

The number b is called the base of the of place-value system of numeration. The known process of counting by different size pebbles or by the use of abacus constitutes a convincing argument. We will not provide a rigorous proof of the theorem. A reader seeking a proof can find it in any treatise in Modern Algebra. We will only illustrate the theorem by different instances of b. Modern society uses, for daily purposes, ten as the base for numeration. The system is called the decimal place-value system. Its use is so implanted in modern minds that it is taken for granted, and a great deal of effort is required to convince most people of the existence of other possible systems. From the mathematical point of view, any system having a base greater than one is just as good as the decimal system. However, some systems are more convenient than others for certain purposes. The only reason for our modern cultures in preferring or ancestors. They are implanted in modern culture that it is impossible to change. However, if I were to rename them, I would suggest “Counting” and “Linear” respectively.

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adopting the decimal system is physiological accident that we have fingers. In the appendix, we will give illustrative examples for the binary and duodecimal systems.

the ten two the

SECTION 3. BINARY SYSTEM The binary system uses only two symbols, which are usually denoted by the familiar symbols for zero and one. For example: 2005 = 1b10 + 1b9 + 1b8 + 1b7 + 1b6 + 0b5 +

1b4 + 0b3 + 1b2 + 0b + 1b0

It is usual to write the above summation by dropping all the bâ&#x20AC;&#x2122;s out, when it is understood from context that b = 2. We in this book, in order not to mix things up we will adopt the convention that the two digits of the system will be denoted by an over bar, namely: 0 and 1. Thus, we get the simple form: 2005 = 11 111 010 101 The following table gives further examples: ! !

!! !

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101

110

100

111

! 8 = 1 000

!9 =

1 001

! 10 = 1 010

!16 = 10 000

!32 =

100 000

! 64 = 1 000 000

!128 ! = 10 000 000

! ! = 256

100 000 000

!512 ! = 1 000 000 000

! !

1024

10 000 000 000

! !

10 000

10 011 100 010 000 !

Table1 (Appendix)

The binary system is the simplest possible system. Its very simplicity makes it convenient in the design of things where there is “off” and “on”, for example in electric circuits, computers, and in some mathematical proofs. However, the lengthy sequences of zeros and ones in expressing even a “small” number, render the binary system cumbersome, awkward, for daily use. The following are the complete tables for addition and multiplication (which do not give head aches to grade school students.)

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!! ! ! !

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Addition

Multiplication

! ! ! !

!1 1 !

! ! !

Table 2 (Appendix) Table of addition ! ! ! ! ! multiplication in the binary system

and

The algorithmic processes for performing the operations of addition, subtraction, multiplication, and division are carried in a manner similar to the usual processes on the familiar decimal system keeping in mind the above table. However, they are not simple to follow due to the lengthy sequences of the two digits. The following examples provide illustrations for processes of the algorithms for the operations of additions and multiplication in the binary system.

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10 100 110

Add

10 111 011

110 101

Multiply

101 1

! !

! ! !

! !

! ! ! ! ! ! ! *

Figure1 (Appendix) multiplication in binary system

! ! ! ! ! ! 1 1 0 1

! ! ! 1 0 1

! ! ! ! ! ! 1 0 1 0 1 ! ! ! ! ! ! 1

! !1 ! !1 !1 ! ! ! Sum 0 0 0 0

! ! 1 ! ! 1 ! !1 Product 0 0

Examples of addition and

! ! ! ! ! ! ! ! !

The following table provides some reciprocals of few numbers. The subscript r stands for recurring fraction.

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1/2

1/4

1/3 = 0 .1 1/5 = 0 . 01 1/6 = = 0 . 100 ( 01)r 1/7 = 1/8 = = 0 . 001 1/9 = ! ! ! ! 1/16 = = 0 . 000 1 1/32 = ! ! ! ! ! ! ! !! ! ! ! !! ! !! !! ! ! !! ! !! ! ! ! As a further example we have: 2 = 10 app 1. 001 010 100 001 . The reader can try =

1/ 10 1/ 100 1/ 110 110 1/ 1 000 ! ! 1/ 10 000 !!

= =

his hand by squaring the above numeral and finding how good the product approximate 10 .

! !

SECTION 4. DUODECIMAL SYSTEM The duodecimal system of numeration uses twelve symbols, which we in this book denote: 0 , 1, 2 , 3, 4 , 5 , 6 , 7 , 8 , 9 , " , " . Where, 2 = 1 + 1, 3 = 2 + 1, 4 = 3 + 1,…, 9 = 8 + 1, " and " stand for 9 + 1 and " + 1 respectively. ! !! ! ! ! ! ! !! ! ! In this section, the “under-barred” digits stand for numbers expressed in the ! ! ! ! ! !! ! ! ! ! !! duodecimal system while the “un-barred” ! ! numbers ! !in the decimal system. ! digits express Thus, 10, 11, 12, 144, and 2005 are written in the duodecimal system as " , " , 10 , 100 , and

! ! ! !

254

1/ 11 1/ 101 1/ 111 1/ 1 001 ! ! 1/ 100 ! 000 ! ! ! ! ! !!

= = = = =

! !

0 .( 01)r 0 .( 001 )r 0 .( 100 )r 0 .( 000 111 )r 0 . 000 01

- 255 -

1 1 " 1 respectively. The reader can easily do

!!! !

all arithmetical manipulations after a little practice. The reciprocals of some numbers are given below (where the subscript r stands again for recurring numeral): 1/2

= 6/b

= 0.6

1/3

= 4/b

= 0. 4

1/4

= 3/b

= 0. 3

1/5

= 0 .( 2497 )r

1/6

!! = 2/b

= 0.2

1/7

= (2/b+ 4/b2+ 9/ b3+7/ b4)r (1/b+ 8/b2+ 6/ ! = b3+10/ b4! + 6 3/b5+ 5/ b! )r !

= 0 .1 6

1/9

= 16/b2

1/11

= (1/b)r

1/144

= 1/ b2

!! 1/8

= 18/b2

!! 1/10 = 1/b+ 1/12

1/5 b = 1/b

! !! !! !

= 0 . 1 ( 2497 )r = 0 .1

One ! !notices that the reciprocals of 2, 3, 4, 6, 8, 9 are easy to manipulate, because 12 has the five factors 1, 2, 3, 4 and 6 while 10 has only the three factors 1, 2, and 5. The advocates of the duodecimal system cite this property as an advantage over the decimal system. Moreover, it reminds us of the advantages of the Mesopotamian sexagesimal system.

255

= 0 .( 186" 35 )r

= 0 .1 4 ! ! = 0 .(1)r

! !! ! ! !!

= 0. 0 1

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We leave the construction of the duodecimal addition table as an easy enjoyable exercise for the reader. The following is the table of duodecimal system of multiplication

9 ! 1 0! 1 3! 1 6! 1 9! 1 "! 2 ! 3 2 6! 2 9

3! 6 !

4! 8 ! 1 0! 1 4!! 1 8!! 2 ! 0! 2 ! 4! 2 ! 8! 3 0!! 3 4!! 3 8

# ! 3! 5! " !! 1 3!! 1 8!! 2 1!! 2 ! 6! 2 ! 4 ! 3 9!! 4 2!! 4 7

10 16 2 0 2 6 30 36 4 0 4 6 5 0 5 6 !! !! !! !! !! !! !! !! !!

# ! 3 6!! 4 1!! 4 ! 4!2 ! 8! 5 ! 3 ! 5 "!! 6 5 7!! 1 2!! 1 9!! 2 !

0! 2 ! 8 ! 3 4!! 4 ! 0!4! 8 ! 5! 4! 6 ! 0 ! 6 8!! 7 4 !! 8!! 1 4!! 2 !

3 ! 3 0!! 3 9!! 4 ! 6 !5 ! 3! 6! 0! 6 ! 9 ! 7 6!! 8 3 !! 9!! 1 6!! 2 !

6 ! 3 4!! 4 ! 2!5! 0 !5" 8! 7! 6 ! 8 4!! 9 2 !! "!! 1 8! ! 2! !! 6 ! $ $

9 ! 3 8!! 4 ! 6 !6 ! 5 ! 7! 4! 8 ! 3 ! 9 2!! " 1 7!5! !! #!! 1 "!! 2 ! 0! 3 ! 0!4! 0 !5 ! 0!6! 0!7! 0 ! 8! 0 ! 8 0 !" ! 0 !# 0 !! 10!! 2 ! !

!! !! !! !! !! !! !! !! ! ! ! ! !

256

100

3 (Appendix) Duodecimal !! Figure !! ! !! !! !! !! !!Multiplication ! ! ! ! Table.

- 257 -

This table is almost as easy to memorize as the corresponding table in the decimal system. Few things must look familiar. • Even numbers end by even numerals. • Multiples of 6 remind us of multiples of five in the decimal system; they end either with digits 0 or 6 . • Multiples of 8 end with 4 , 0 , or 8 . ! • Multiples of " begin with the multiplier minus one: like nine in the familiar ! ! decimal system. For example: 5 x " = •

! ! ! ! 47 ! The sum of the digits of any multiples

of " is a multiple of " ; again like nine

! !

in the decimal system.

The following examples provide ! ! illustrations for processes of the operation of additions and multiplication in the duodecimal system. The reader, by utilizing the above table for multiplication and the table for addition, which he was asked to construct, is requested to follow the procedure of the operations and to double check the accuracy of the results by transforming the numbers into the decimal system.

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Add

763 # 5 " 4#1

3# "# 4 39 "

Multiply

52 " 0# 4

#2" 5

7 6

Sum ! !7 ! !# !2

! ! !

# !4 ! 3 !8 ! ! ! ! !

Product

As a further example, we have: 3 = app 1. 895 . The reader can try his hand by squaring the above numeral and finding how close the product approximates 3.

!! It is interesting to mention that there are societies that call for adopting the duodecimal system and ! abandoning the decimal system. They suggest the names “gross” for 100 , “grand” for 1 000 and “twelves” for 1/ 10 . The following are among some of the advantages they cite: ! The number!!of digits for ! ! “large numbers” takes less paper space. The duodecimal system is

258

! ! #

4 4 5 2 ! !* !* !** ! 4 !5 !8 !6

! ! !

Figure 2 (Appendix) Examples of addition and ! ! !in the ! duodecimal ! ! ! ! ! ! ! multiplication system

8 !$ # !8 ! !4

# 1 9 ! ! 1 !8 !7 4 !* !* !# #

! ! ! ! ! ! ! ! !

2 5

- 259 -

more practical in expressing the number of stars in the Universe and to express the USA national debt. The clock digits are easier to read and manipulate. The year is divided into 10 months. A foot is 10 inches. A foot is 0 . 2 5 yard. The Zodiac is!divided into 10 zones. !The digits of written ! !! numerals are usually grouped by 3’s and 4’s; this is why phone numerals are written in the form: (705) 921- 3746 as an example. Moreover, eggs are sold in dozens, that is in 10 ’s. Beer is sold in six packs (never in five packs)!

! information about activities of For more those societies, see Dozenal Journal published in Great Britain.

SECTION 5. PROOF OF PYTHAGOREAN -TRIPLE THEOREM

THE

The Pythagorean-triple theorem was stated in section 09.4. The purpose of this

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appendix is to provide a proof for the theorem. The proof depends mainly on properties of odd, even numbers, and the notion of common factor. The proof, though lengthy, is easy to follow. It does not utilize Mathematical WMD (That is, weapons of mass deduction.) Notes: (1) The word “number” in this section means “a positive integer.” (2) We describe a set of numbers to be pairwise co-prime if there is no factor common to the elements of the set except one. This section is devoted to the proof of the classification theorem of Pythagorean triples. The theorem is preceded by some preparatory remarks.

! ! !

Remarks Let (a,b, c) is a Pythagorean triple. Then: (0) The three numbers a , b and c cannot all be odd. (1) If moreover, the triple is primitive then the set ! { a , b , c} is pairwise co-prime. (2) If moreover, ! !the triple ! is primitive then a , and b cannot be both odd. (3) If moreover, the triple is primitive then no two of the three numbers a , b , c can be even. Proof of Remark (0) 2 2 2 Appealing to the equality c = a + b , the oddness of a , ! and!b ! would force c to be even.

! ! !

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Proof of Remark (1) Pythagoresness of the triple makes any factor common to a pair from the triple to be a factor of the triple, which contradicts the primitiveness of the triple.

Proof of Remark (2) Assume, contrary to the hypothesis, that a , and b are odd. It follows from Remark (0) that c is even. The evenness of c implies that 4 is a factor of c 2 ; but the oddness of a , and b makes a 2 + b 2 of ! equality c 2 the form 4k + 2 for some integer k . The ! 2 2 = a + b implies that 4 is a factor of 2 , which is a ! glaring contradiction.

! ! (3) Proof of Remark Assume, ! contrary to!the hypothesis, that at least two of the three numbers a , b and c are even. This assumption contradicts the pairwise coprimeness of the triple, which in turn, by virtue of (1) contradicts the primitiveness of the triple.

! !

THEOREM A triple

(a,b, c) is primitive Pythagorean if an only if

p and q not both are odd such that. c = p2 + q2 2 2 The odd entry of the triple (a,b, c) = p " q The even entry of the triple (a,b, c) = 2 pq . ! ! ! ! ! ! ! !

There is a pair of coprime positive integers •

• •

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Proof

! !

Let (a,b, c) be a primitive Pythagorean triple. There are eight possible cases according to the three numbers being even or odd. The Remarks (1), (2) and (3) banishes six of the cases, the remaining possibility is either ! or b, a , c are odd and b is even c are odd and a is even. Let us prove the theorem by taking the first mentioned possibility holding true. The proof of the when the !theorem ! ! second possibility is ! true can be performed “manually” ! ! by simply interchanging in the proof, given below, the letters “a” and “b”, wherever they occur. Let us proceed with the proof. Since a , and c are odd and b is even, therefore the three numbers (a + c) / 2 , (c " a) / 2 , and b / 2 are integers. Therefore, there is a sense in talking about their common factor. ! Let d!be a common factor ! of (a + c) / 2 and (c " a) / 2 . It is easy!to see that,!d is also a factor of the numbers a, and c. Consequently by, Remark (1), d has to be one. Hence, there is no prime factor ! !, and (c " a) / 2 . On the common to both of (a + c) / 2 other hand, we have ! (b 2)2 = (c 2 " a 2 ) 4 = ( (a + c) / 2 ) x( (c " a) / 2 ). This means that the product of (a + c) / 2 !a) / 2 is a perfect ! with (c " square. This cannot happen unless each prime factor of the two numbers ! even exponent. !(a + c) / 2 and (c ! " a) / 2 , has an This leads that each !of (a + c) / 2 and (c " a) / 2 is a perfect square. Let r and s be their

! ! !

262

Remember The sum of two odd numbers is even. & The difference of two odd numbers is even.

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! ! !

square roots respectively. Hence b 2 = r s . Note that r has to be larger than s and that r " s = a . 2 2 Solving for a , c we obtain a = r " s and 2 2 c = r +s . !! Note that, since! a being odd and equals ! r and ! s cannot both (r + s)(r " s) , then ! the numbers ! ! be odd. ! ! This completes the proof of the “ if ” part of the theorem.!

Now, let us prove the converse, that is the “only if ” part. Assume, p and q , are two positive integers that satisfy the conditions given by the theorem, namely they are two co-prime numbers, one of which, say, p is odd; therefore, q is even. We will show that 2

the triple ! ( p! " q , 2 pq , p Pythagorean triple. Let the letters x ,

+ q 2 ) is a primitive

y , and z stand for p 2 " q 2 , 2 pq , and p 2 + q 2 p 2 + q 2 respectively. ! means!we have ! to show that the triple (x, y, z) This is a Pythagorean ! triple. !2 2 Consider x + y =

!2 2 2 ( p " q ) + (2 pq)2 ! ! 4 2 2 4 2 2 =(p " 2 p q + q )+ 4 p q ! 4 2 2 4 = p +2p q +q ! ! 2 2 2 =(p + q ) 2 =z ! Thus, the triple (x, y, z) is Pythagorean. In ! order to complete the proof, it remains to show that it ! is primitive. ! ! !

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Let, k be a common factor for x , and z . Since, p is odd and q is even, it follows that each of x and z is odd. The factor k , has to be odd, and has to divide both x + z and z " x . Consequently, k ! divides factor, k! being odd 2 p 2 and 2q 2 . The ! ! cannot be ! two. Hence, k must be a factor of both p ! ! and q . ! q are co! ! assumption, p , and However, by ! ! prime, forces ! k to be one.

This completes the proof that the triple (x, y, z) is primitive Pythagorean. ! !

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Appendix

(History) This appendix exhibits a brief global “time-line” of great events in Mesopotamia such as the rise and fall of empires, wars, conquests, defeats, revolts, destruction of cities and some other “parallel” important events happening elsewhere.

Historical Event

Ur dominated Sumer. Sargon I The Great establishing the Akkadian Empire and dominating over Mesopotamia.

Approxi mate

Parallel

Period BC.

Events

? – 2600

Cuneiform writing was developed.

2600 - 2200

Epic of Gilgamesh was written.

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Assyrians settling in the North of Mesopotamia. Guti overrunning Sumeria Ur restoration.

2200-1800

Old Assyrian Empire establishment.

1800-1600

Hammurabiâ&#x20AC;&#x2122;s reigning over Babylon.

1792 - 1750

Hittites invading Syria. Hittites pillaging Babylon.

Laws were codified.

1600-1500 1600 - 1500

Kassite seizing power in Babylon. 1500 -1400

The first historically recorded battle at Megiddo in Palestine (Armageddon).

Establishing Middle Assyrian Empire.

1400 -1300

Akhenaton introduces monotheism in Egypt.

Assyria expanding into Syria.

1300 -1200

Fall of troy.

Mittani dominating Assyria. Kassites conquering Sumer

Elemites seizing control of Babylon. Nebuchanzzar I restoring native Babylon.

1200 -1100

Arameans defeating Assyria. Chaldeans establishing empire in southern Mesopotamia.

Assyria expansion renewal.

266

1100 -1000

1000 - 900

Phoenicians (known in the Bible as Canaanites) develop alphabets.

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Establishing new Assyrian Empire.

900 - 800

Assyria conquers Syria Cimmerians attacking Assyria

800 -750

Assyria annexing Syria, conquering Babylon and Palestine.

750 -700

Assyria defeating Cimmerians and conquering Egypt. Chaldeans expanding north toward Babylon.

Founding of Rome. Deportation of the ten tribes of Israel.

700-650 700-650

Assyria conquering Elam. 646

Assyrians are ejected from Egypt.

Egypt invading Syria.

Later Babylonians defeating Egyptians. Chaldeans revolting against Assyria.

625

Chaldeans seizing power in Babylon. Chaldeans conquering Syria and Elam.

625 - 612

Destruction of Nineveh.

612

Destruction of Assyria.

608

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268

Chaldeans invading Palestine.

597

Chaldeans destroying Jerusalem.

586

Persia conquering Elam.

556

Persia conquering Assyria.

547

Persia conquering Babylon.

539

Persia conquering Egypt.

525

COLLAPSE OF MESOPOTAMIA EMPIRE.

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References

269

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J.H. Breasted

“A History of Ancient Egyptians”., Peter Bredrick Books 1999.

[Br]

E. M. Bruins

On Plimpton 322, Pythagorean Numbers in Babylonian Mathematics. Afdeling Naturkunde, Proc. 57 (149),629632.Akad.v. Wetenshoppen, Amsterdam.

[Bu]

R.C. Buck

Sherlock Holmes i n Babylon. Amer. Math. Monthly 87 (1980)335-345

[B]

[D]

Dozenal Reviews. H. Eves

“An Introduction to the History of Saunders College Publishings,1969.

[Ha]

G. Hardegree

“Numeration Systems” University of Massachusetts March 1, 2001.

[Ho]

L. Hogben

“Mathematics for the Million” WW Norton & Co. New York, London, 1993.

[Ka]

R. Kaplan

“The Nothing that is: A Natural History of Zero” Oxford, New York,1999.

[Kr]

S. N. Kramer

“Cradle of Civilization.”

[N]

O. Neugebauer

The Exact Sciences in Antiquity”. Second edition,1957 Brown University Press, Dover reprint,1969.

O. Neugebauer and

Mathematical Cuneiform Texts. Amer. Oriental Series 29. American Oriental Society, New Haven,1945.

[E]

[NS]

A. Sachs [OR 1]

J. J. O’Connor, and E. F. Robertson

Pythagoras’s Theorem in Babylonian Mathematics, http://www.groups.dcs.st.and.ac.uk/~histTopics/Babylo nianPythagoras.html.2000.

J. J. O’Connor, and

Indian Numerals

E. F. Robertson

http://wwwhistory.mcs.standrews.ac.uk/HistTopics/Indian_numera ls.htm

J. J. O’Connor, and

A history of Zero

E. F. Robertson

http://www-history.mc.standrews.ac.uk/HistTopics/Zero.html

[R]

G. Roux

“ Ancient Iraq” 3 rd edition Press,1998.

[Z]

E.C. Zeeman

http://www.Math.utsa.edu/ecz

[OR 2]

[OR 3]

270

Mathematics”,

Princeton

University

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271

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Index

272

11/22/06 . Numerals ..................................................... astrolabe .......................................................136 26 . Robertson ..................................................... Astronomical51Diaries................................... 135 “ atomic” numbers......................................... atomic ............................................................. 37 37 “Al-Khawarizmi on axiomatic method......................................... 121 the Hindi Art of axioms........................................................... 121 Reckoning ................................................. 70................................................ 37 Baby numbers “baby numerals” of baby numerals ................................................ 37 the system.................................................. 43 Babylon ............................................................ 9 “can-opener .................................................... Babylonian Flood 32 “Number-gate” ............................................. story........................................................... 65 22 Babylonian hanging “Numeration gardens...................................................... 61 systems...................................................... 27 “Ya Aleel, Ya Anu........................................ Baghdad............................................................ 125 8 abaci ............................................................... base of the system 32 .......................................... 43 abstract notion of Basra ................................................................. 8 numbers.................................................... Beauty ............................................................ 33 26 abstraction ...................................................... Behistun.......................................................... 90 24 addition is binary operation ............................................. 91 commutative............................................ Brahmagupta91(598 Al-Biruni (973AD – 670 AD) .......................................... 69 1048). ........................................................ Bruins, E.M.................................................... 70 51 Aleel .......................................................50, Cain .................................................................. 125 4 Alexander the Great ...................................... Canaanites ...................................................... 67 21 Alexander the Great canned” solution ............................................ 31 (356 BC – 323 Cardan ............................................................ 72 BC)............................................................ cardinal numbers............................................ 61 28 Alexandria...................................................... Chaldians.................................................... 66 9, 13 algebra ....................iv, 62, 71, 91, Champollion................................................... 92, 122, 128 23 algorithm ........................................................ China ..................................... 70 ii, 4, 33, 122, 124 Al-Khawarizmi city-states ......................................................... 6 classification (about 780 AD – theorems.................................................. 114 about 850 AD).......................................... 70 Anatolia .......................................................... compound interest 20 ......................................... 76 Anu .........................................................50, computer languages....................................... 125 94 Arabic-Islamic computer program ......................................... 94 civilization ...........................................iii, computer software 70 Aramaic ............................................................ icons .......................................................... 9 18 Archimedes.............................................43, conceptual121 leap in Armenia.......................................................... abstraction 20................................................ 36 Assurbanipal .................................................. conjecture. ..................................................... 22 34 Assyrians,......................................................... cubic equation9 ................................................ 96

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cultural vacuum ............................................. Irbeel................................................................. 36 9 Darius ............................................................. Justice ............................................................ 24 26 Diophantanus (about Karl Popper .................................................... 51 200 AD – 284 Kepler ........................................................... 135 AD), ........................................................ Kidinnu (? 114 – 375 BC ................................... 136 Diophantus ............................................... King Cyrus v, 123 ..................................................... 12 dozen .......................................................... King Cyrus ii, II 52................................................. 61 Egyptii, v, 4, 13, 23, 24, 38, 62,King 66, 108, Sennacherib 114, 122,.......................................... 124, 152, 153 22 Egyptian Kirkuk............................................................... 9 Hieroglyphic............................................. Lagash .............................................................. 23 9 empty set......................................................... Laplace (1749 57 – Epoch of Gilgamesh. ..................................... 1827) ......................................................... 21 43 Eridu ................................................................. Lascaux ............................................................ 9 2 Euclid ...................................................121, Leonardo Da 129Vinci .......................................... 2 Euclidean criteria ..................................96, Love ................................................................ 120 26 Euphrates.......................................................... Maya ............................................................... 8 73 Fibonacci, Leonardo Mayan.....................iv, 1, 28, 39, 43, 51, 72, 73 (1170 - 1250)............................................ Mesopotamia................................................ 71 ii, 8 Mesopotamian Fourier (1768- 1813) ..................................... 23 civilizations......................... iii, 1, 13, 15, 50 French numberwords......................................................... Mesopotamiansii, 52 iv, vi, vii, 12, 13, 14, 15, 17, 18, 19, 21, 22, 24, 28, 36, 38, 39, 45, 46, 4 Galileo .......................................................... Mexico............................................................ 135 73 Galois Theory ................................................ Middle Stone-Age 66 ........................................... 3 Gardegree, Gary ............................................ million ............................................................ 59 31 Great Agricultural Mosul.......................................................... 9, 21 Revolution .................................................. Nabu-rimannu3 (? – Greek Miracle .................................................. 490 BC)................................................... ii 136 Greek Number-Gate ...................................... naked ape.......................................................... 61 2 Guatemala ...................................................... 73O. ......................................... 112 Neugeauber, hairy ape ........................................................... 2 Neugebauer, Otto........................................... 51 Homo sapiens................................................... 1 Newton ......................................................... 135 hybrid system ................................................. 47 Nineveh ........................................9, 21, 22, 153 inadequacy of the Nippur............................................................... 9 symbols used............................................. 45 ........................................... 28 nominal numbers India.................... ii, 4, 33, 67, 70, 71, 122, 124.............................................iv, 65 Number-gate infinite ............................................................ 27 Numbers......................................................... 26 infinite set....................................................... 38 numerical analysts ......................................... 77 Iran............................................................ 8, 110 ................................................. 1 Old Stone-Age Iraq.................................ii, 8, 9, 12, 14,Al 22,Khayyam 103 Omar ........................................ 15 Iraq-Iran war (1980 – one-to-one 1988). .......................................................... 8 correspondence ........................................ 30

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ordinal numbers ............................................. Santayana ......................................................... 28 ii Original sexagesimal point. ......................................... 47 Mesopotamians........................................... Shatt-Al-Arab9 .................................................. 8 Palestine. ........................................................ Sherlock Holmes 59 ..................................112, 124 Persia .............................................................. slide ruler........................................................ 20 48 Phoenicians .................................................... Strabo of Amasia 21 (64 phonetic alphabet........................................... BC – 23 AC)........................................... 21 136 Picassos ............................................................ Suliamnia ......................................................... 3 9 pictograms ..................................................... surds................................................................ 17 65 place-system of Swerdlow ..................................................... 135 numeration................................................ Syria......................................................8, 39 20, 59 place-value tablet. .............................................................. 17 numeration tally sticks, ..................................................... 36 system........................................................ Thales ............................................................. 27 62 place-value system ......................................... Thales of Miletus iv ......................................... 122 The Babylonian Plimpton 322........................... v, 115, 116, 117 Theory of the Poincare............................................................ ii Planets..................................................... 135 Pope Sylvester ............................................... 71 postulates ..................................................... Tigris ................................................................ 121 8 primitive trigonometry................................................. 104 truncated Pythagorean sexagesimal triple. .......................................................... vi system.”103 ..................................................... 46 Prophet Abraham......................................... psycho-philosophical truncated mental complex ........................................ sexagesimal 57 Ptolemy in Almagest...................................... system: ...................................................... 66 44 Pythagoras (about Turkey .............................................................. 8 569 BC – about undefined notions. ...................................... 121 understanding 475 BC)..................................................... 62 from context”122 ..................................................... 46 Pythagorean school ..................................... Pythagorean theorem.... 99, 108,Ur 123, 9, 103 124, 127 urbanization Pythagorean triple....................................... 113 revolution.................................................. 16 Pythagorean triples. v, 109, 114, 115, 116, 148 Rather be ambiguous Venus Figurines ............................................... 2 than fallacious .......................................... 58 vigesimal system............................................ 51 Rawlinson, Sir Henry Ya lail. Ya ain ................................................ 50 (1810 - 1895), ........................................... Zeeman E. C. 23...............................................119 Robertson E.F. ............................................... ziggurat........................................ 51 104, 125, 127 Roman system................................................ Ziggurat. .......................................................103 27 Rousseau .......................................................... zodiac ............................................................. 1 50 π 107 Sachs............................................................ 112

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