............................................................
the encyclopedia of
numbers
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contents
introduction
zero…beyond one million…infinity
numeral systems
tally…mayan…roman…alpha-numeric
evolution of the glyphs
evolution…hindu-arabic…abacus to numerals
classification of numbers
cardinal…number lines…natural…rational
pi pi and circles…piphilology…grouping numbers
the fibonacci sequence
the spiral…in the human body…in nature
numerology
core numbers…definitions…master…sun
lucky and unlucky numbers
across the world…lucky…unlucky…thirteen
numbers in language one to five…six to nine
bibliography
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contents
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history mathematics ..................................................................... .................................................
an introduction to numbers ..................................
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culture ...................................... ..............................................................................
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an introduction to numbers
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Infinity is not a number, it is bigger than the biggest number. It is an idea. Infinity is the state of being without finish. ...................................................................................................
Numbers are infinite, therefore gaining a full understanding is impossible. To give some structure to the subject, this chapter gives an overview of the following numbers: • • •
Zero Beyond one million Infinity
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introduction ...................................................................................................
an introduction to numbers
zero ................................................................................................... How can nothing be a number?
The place holder system
Zero is a very strange number. It is neither positive or negative. If you add or subtract zero to any number, that number stays the same. If you multiply any number by zero, you get zero. Any number raised to the power of zero is one, so 20=1.You cannot divide a number by zero.You cannot take the zeroth root of a number. It is difficult to say what zero is. Zero seems normal today, but it confused the ancient Greeks, and medieval people. How can nothing be a number? When arabic numbers were accepted, the meaning of the number zero became understood. It would be impossible to have modern mathematics without it.
The other meaning of zero is as a place holder. Early number systems had no such thing as a number zero. Number systems will be explored in more detail in chapter two, however, in the case of zero, it was the place-holder system that made zero a relevant digit. This used the position of the numbers to show how big they were. Today’s system works in the same way. For example, 123 is bigger than 97, although the digits 9 and 7 are bigger than 1, 2 or 3. Numbers are put into columns: units, tens, hundreds, etc. This means that there may be no digit in one particular column, such as two hundreds, no tens and five units. This is written as 205. The middle zero is a place holder in an empty column. This system gave zero a consistent place in mathematics.
10, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, etc.
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googolplexian: the first 1, 939 digits ...................................................................................................
an introduction to numbers
infinity ..............................................................................................................
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Infinity is an abstract concept describing something without any limit. It is relevant in a number of fields, predominantly
mathematics and physics. The word infinity derives from the meaning ‘the state of being without finish’.
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history mathematics ..................................................................... .................................................
numeral systems ..................................................
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numeral systems
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A numeral system is a set of symbols or mathematical notations that are used in a consistent manner to represent numbers of a given set.
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It can be seen as the context that allows the symbols ‘11’ to be interpreted as the binary symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases. Ideally, a numeral system will: • • •
Represent a useful set of numbers (e.g. all integers, or rational numbers) Give every number represented a unique or standard representation Reflect the algebraic and arithmetic structure of the numbers.
This chapter will explore the following numeral systems: • • • •
tally mayan roman alpha-numeric
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introduction ...................................................................................................
numeral systems
tally marks .............................................................................................................. The tally system is arguably the simplest numeral system ever used. It makes use of single strokes to represent objects being counted. Tally marks are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate results need to be erased or discarded. However, because of the length of large numbers, tallies are not commonly used for static text. Although one advantage is simplicity, there is a disadvantage. Large numbers require alot of strokes and become difficult to read.
One stroke is used to count each object as follows:
1
2
3
As the number gets larger, the becomes more difficult to read:
In this case, a fifth tally is marked across every four to make a group of five::
Tally marks used in most of Europe, Turkey, Zimbabwe, Australia, New Zealand and North America.
Cultures using Chinese characters tally by forming the character which consists of five strokes.
Tally marks used in France, Spain, South America (Argentina, Brazil, Chile, Venezuela and Uruguay, among others) and French-speaking Africa. In Spanish countries, these are most commonly used for registering scores in card games.
In the dot and line tally, dots represent counts one to four, lines five to eight, and diagonal lines 9 and 10. This method is commonly used in the forestry field.
Different cultures around the globe use different versions of the tally system. However, overall it is a very basic numeral system used for simple counting and record-keeping only.
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numeral systems
mayan numerals ..............................................................................................................
The table on the right shows the mayan numerals one to twenty. Dots represent numbers one to four and a bar represents five. The shell symbol can represent a zero, or multiples of ten. In this case, the shell represents twenty.
5
+
8
+ 6
+ +
=
13
= 4
= =
10
Note that in the second addition example, the six plus four would give five dots. In this case, the dots are replaced by a bar.
The mayan number system was developed by the ancient naya civilization of Central America. They used the four fundamental operations: addition, subtraction, multiplication and division. It is believed that calculations were made on the ground or even surfaces using stones of branches as writing tools. To accomplish mathematical operations they used a table where the points and bars were placed. Similar to the number system we use today, the mayan system operated with place values. To achieve this place value system they developed the idea of a zero place-holder. The Maya seem to be the first people who used a place value system and a symbol for zero. Beyond these similarities there are some significant differences
between the mayan number system and our modern system. The mayan system is in base-twenty (vigesimal) rather than base-ten (decimal). The numerals are made up of three symbols: zero (shell shape), one (a dot) and five (a bar). For example, thirteen is written as three dots in a horizontal row above two horizontal lines stacked above each other. The two important markers in this system, five and twenty, are thought to relate to the fingers and toes, twenty in total, and five digits on each hand and foot. Addition and subtraction: Adding and subtracting numbers below twenty using maya numerals is very simple. Addition is performed by combining the numeric symbols at each level.
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numeral systems
alpha-numeric system ..............................................................................................................
*theta, koppa and sampi are obselete
The alpha-numeric system uses letters from the greek alphabet. These alphabetic numerals are also known by names Ionic or Ionian numerals, Milesian numerals, and Alexandrian numerals. In modern Greece, they are still used for ordinal numbers and used similarly to Roman numerals in the West. Greek numerals are decimal, based on powers of ten. The units from one to nine are assigned to the first nine letters of the old Ionic alphabet from alpha to theta. Instead of reusing these numbers to form multiples of the higher powers of ten, however, each multiple of ten from ten to ninety was assigned its own separate letter from the next nine letters of the Ionic alphabet from iota to koppa. Each multiple of one hundred from one hundred to nine hundred was then
assigned its own separate letter as well, from rho to sampi. Unfortunately, this method of counting needs twenty-seven letters, and there were only twentyfour in the classical Greek alphabet. This meant that the Greek’s had to find three extra symbols for the missing numbers of six, ninety and nine hundred. They used three archaic letters, which used to be in the alphabet but had been dropped as they were no longer required (theta, koppa and sampi). In ancient and medieval manuscripts, an over bar was used to distinguish these numerals from letters. Over time this changed to a mark on the upper right of the numeral.
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numeral systems
roman numerals ..............................................................................................................
I II III IV V VI VII
I
1
V
5
X
10
L
50
C
100
D
500
M
1000
X
10
LX
60
XX
20
LXX
70
XXX
30
LXXX
80
XL
40
XC
90
L
50
C
100
C
100 DC
600
CC
200 DCC
700
CCC
300 DCCC
800
CD
400 CM
900
D
500 M
1000
The tables of roman numerals show how the seven symbols are used for numbers up to one thousand. Numbers up to five thousand could be expressed using the seven symbols, above this special considerations had to be devised.
I VIII IX X XI XII
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in a few specific cases, to avoid four characters being repeated in succession (such as IIII or XXXX), these can be reduced using subtractive notation: ...
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The roman numeral system is based on seven symbols, shown in the far left table. Use of Roman numerals continued after the decline of the Roman Empire. From the 14th century on, they began to be replaced in most contexts by more convenient Hindu-Arabic numerals. However, this process was gradual and the use of Roman numerals in some minor applications continues to this day. Numbers are formed by combining symbols together and adding the values. So II is two ones, (or 2), and XIII is a ten and three ones, (or13). There is no zero in this system, so two hundred and seven for example, is CCVII, using the symbols for two hundreds, a five and two ones. 1066 is MLXVI, one thousand, fifty and ten, a five and a one. Symbols are placed from left to right in order of value, starting with the largest. However,
..........
I can be placed before V and X to make 4 (IV) and 9 (IX). X can be placed before L and C to make 40 (XL) and 90 (XC). C can be placed before D and M to make 400 (CD) and 900 (CM).
...
Although Roman numerals were replaced by Hindu-Arabic numerals, they remained in many niche contexts such as clock faces, monarch names, book volumes, etc.
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history mathematics ..................................................................... .................................................
evolution of number gly
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yphs ..................................................................................... chapter ...
evolution of the glyphs
introduction ..............................................................................................................
‘It is India that gave us the ingenuous by the means of ten symbols, each sy as well as an absolute value; a profou appears so simple to us now that we simplicity, the great ease which it has arithmetic in the first rank of useful i the grandeur of this achievement wh the genius of Archimedes and Apollo produced by antiquity.’
s method of expressing all numbers ymbol receiving a value of position, und and important idea which e ignore its true merit, but its very s lent to all computations, puts our inventions, and we shall appreciate hen we remember that it escaped onius, two of the greatest minds
The significance of the development of the positional number system is described by the French mathematician Pierre Simon Laplace (1749–1827)
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evolution of the glyphs
evolution of the glyphs ..............................................................................................................
West Arabic, circa. 11th century
15th century
16th century
Indian, 1st century AD
Indian, 9th century
East Arabic, circa. 12th century
Indian, circa. 11th century
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evolution of the glyphs
the hindu-arabic system ..............................................................................................................
place value
digits
equals
thousands
hundreds
tens
ones
4
6
8
7
4x1000
+
6x100
+
8x10
+
7x1
4000
+
600
+
80
+
7 4687
123 can be represented as + – – ///
......
This is called sign-value notation. Tally marks and mayan numerals discussed in chapter two are numeral systems of this type. More elegant is a positional system,
304 = 3×100 + 0×10 + 4×1 ........................................... ...
.. 304 can be represented as +++ ////
also known as place-value. Again working in base-ten, the ten digits zero to nine are used and the position of a digit signifies the power of ten that the digit is to be multiplied with. For instance,
.....
/ stands for 1 – for 10 + for 100
.....................
The simplest numeral system is the unary numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the number seven would be represented by ///////. Tally marks represent one such system still in common use. The unary notation can be abbreviated by introducing different symbols for certain new values.Very commonly, these values are powers of ten. For instance, ......
(Note that zero, which is not needed in the other systems, is of crucial importance here, in order to be able to ‘skip’ a power).
Aryabhata of Kusumapura developed the place-value notation in the 5th century and Brahmagupta later introduced the symbol for zero. The numeral system slowly spread to surrounding countries and it was adopted and developed by the Arabs. The Arabs translated Hindu texts on numerology and the Western world modified them and called them
the Arabic numerals, and hence the current Western numeral system is the modified version. This system is widely used around the world today, and can be summarised by its four main attributes. 1. It uses ten digits that can be used in combination to represent all possible numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. 2. Ten ones are replaced by one ten, ten tens are replaced by one hundred, one hundred is replaced by one thousand, etc. 3. It uses a place-value starting from right to left. The first number represents how many ones there are. The second number represents how many tens there are, etc. 4. The system is additive. The value of a numeral is found by multiplying each place value by its corresponding digit and then adding the resulting products.
Modern day keyboards and keypads in arabic countries still display both the Western Arabic numerals and the ArabicIndo numerals. For example,
1
2
3
4
5
6
7
8
9
*
0
#
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evolution of the glyphs
The woodcuts show keen competition between the algorist and the abacist. The smiling man has discovered Hindu-Arabic numbers; the other still uses the abacus.
The moved to the Hindu-Arabic numeral system was slow. The great beauty of the abacus was that it did not rely on a numeral system. Regardless of whether one could read or write, it can be used to solve most practical numerical problems, which means that even uneducated merchants or traders could carry out the kinds of mathematical transactions involved in business, from keeping accounts to calculating interest. As a result, the abacus became one of the sine qua nons of the Western world, a commonplace and indispensable tool until the adoption of Hindu-Arabic numbers and the gradual spread of numeracy and literacy led to its extinction. While it may seem that the abacus is an ancient tool, it was widely used until a few hundred years ago.
Hindu-Arabic math entered Europe with the great Moorish invasions of the eighth and ninth centuries, and it spread with snail-like slowness. Depending on the region, the transition to HinduArabic numerals occurred between the thirteenth and seventeenth century. They appeared first in Italy and Spain, which, being on the Mediterranean, were closest to the Arab world, and much later in France, England, and Germany. The switch also occurred in different social classes at different times, with the educated upper classes learning the new notation long before the unlettered lower ones. In general, the Hindu-Arabic system was commonly employed throughout Europe by the end of the sixteenth century.
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from abacus to hindu-arabic numerals ...................................................................................................
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classification of n
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culture ...................................... ..............................................................................
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numbers .............................................................................................
classification of numbers
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Different types of numbers are used in many cases. Numbers can be classified into sets, called number systems*. ...................................................................................................
*note these are different to numeral systems as described in chapter two.
There are many different classification systems, this chapter will give an overview of these systems, and focus more specifically on the the most important and common classifications. These include: • • • •
Cardinal/Ordinal/Nominal Number Lines Natural/Integers/Rational/Irrational Famous Irrational Numbers
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introduction ...................................................................................................
classification numeral of numbers systems
Cardinal Numbers
Ordinal Numbers
Nominal Numbers
A cardinal number is a number that says how many of something there are, such as one, two, three, four, five.A cardinal number answers the question ‘How Many?’ It does not have fractions or decimals, as it is only used for counting. An easy way to remember this is, ‘Cardinal is Counting’.
An ordinal number is a number that tells the position of something in a list, such as first, second, third, etc. An example of this is shown above. The highlighted man is sixth. An easy way to remember this is to say, ‘Ordinal numbers are the Order things are in.’
A nominal number is a number used only as a name, or to identify something (not as an actual value or position). For example, the number on the back of a footballer’s shirt is nominal, as is a postal code. An easy way to remember this nominal numbers is to say, ‘Nominal is a Name.’
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classification of numbers ...................................................................................................
classification of numbers
classification of numbers ..............................................................................................................
-121/6
-5.4
-2.984
-63/4
-1.5
-2.984
Number Lines A number line is a horizontal straight line that has points, equally spaced, which correspond to specific numbers. Each number is greater to all the numbers on its left and less than all the numbers on its right. A whole number line is the simplest, it includes only numbers above zero, (i.e. 1, 2, 3, 4, 5, etc. ) An integer number line included positive and negative numbers or ‘integers’ (i.e. -3, -2, -1, 0, 1, 2, 3). The third type of number line includes fractional or decimal numbers. Fractions or decimals are put between the whole numbers, at intervals that reflect the size of the number at a given point. For example, ž is placed three quarters of the way between zero and one.
The number line on this page is an almalgamation of all three types of number line in one. The variation of numbers in a number line show there are many different ways to classify numbers. The following pages in this chapter will briefly explore some of the more complex classification systems, used mainly by mathematicians.
5.25
2.5
...................... ...................................
33/5
...
101/2
7.245
4.09
9.9
4.25
4.125
4.5 4.33
4.75
4.45
4.5698
4.875
4.79
4.94
....
...
4.184
4.451 4.30
4.523
4.64
4.54286742
4.85
4.93 4.921
The above number line magnifies the line between numbers four and five, demonstrating the infinite nature of whole, decimal and fractional numbers.
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classification of numbers
classification of numbers ..............................................................................................................
N
Natural
0, 1, 2, 3, 4, ... or 1, 2, 3, 4
Z
Integers
-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5
Q
Rational
a/b
R
Irrational
�, �
(where a and b are integers and b is not zero)
Integers
The most familiar numbers are the natural numbers or counting numbers: 1, 2, 3, and so on. Traditionally, the sequence of natural numbers started with one (zero was not even considered a number for the Ancient Greeks.) However, in the 19th century, set theorists and other mathematicians started including zero in the set of natural numbers. Today, different mathematicians use the term to describe both sets, including zero or not. The mathematical symbol for the set of all natural numbers is N. In the base-ten numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
An integer is a number that can be written without a fractional or decimal component. For example, ........
Natural numbers
..... 21, 4, and −2048 are integers
9.75, 5½, and √2 are not integers.
The negative of a positive integer is defined as a number that produces zero when it is added to the corresponding positive integer. Negative numbers are usually written with a negative sign (a minus sign). As an example, ......
....
the negative of 7 is −7, and 7 + (−7) = 0.
Irrational Numbers
A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero integer denominator. In the fraction written m/n or m represents equal parts, where n equal parts of that size make up m wholes. Two different fractions may correspond to the same rational number. For example,
If a real number cannot be written as a fraction of two integers, it is called irrational. A decimal that can be written as a fraction either ends (terminates) or forever repeats, because it is the answer to a problem in division. Thus the real number 0.5 can be written as ½ and the real number 0.333... (forever repeating 3s, otherwise written 0.3) can be written as ¹/3. On the other hand, the real number � (pi), the ratio of the circumference of any circle to its diameter, is � = 3.14159265358979... Since the decimal neither ends nor forever repeats, it cannot be written as a fraction, and is an example of an irrational number.
.....
Rational Numbers
.........1/2 = 2/4 = 3/6 = 4/8 If the absolute value of m is greater than n, then the absolute value of the fraction is greater than one. Fractions can be greater than, less than, or equal to one and can also be positive, negative, or zero. The set of all rational numbers includes the integers, since every integer can be written as a fraction with denominator one.
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classification of numbers
�
Pi* as a famous irrational number. People have calculated pi to over a quadrillion places and still there is no pattern. The first few digits look like this: 3.1415926635897932384626433832795…
e
The number e (Euler’s number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. The first few digits look like this: 2.7182811828459…
ф
The golden ratio is an irrational number. The first few digits look like this: 1.61803398874989484820…
�
Many square roots, cube roots, etc. are also irrational numbers. An example is: �3 = 1.7320508075688772935274463415059… *Pi is arguably the most well-known irrational numbers and is often represented in popular culture. For this reason, chapter five will explore the subject of Pi.
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famous irrational numbers ...................................................................................................
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pi
introduction ..............................................................................................................
3.14159265358979323846264338327 944592307816406286208998628034 664709384460955058223172535940 055596446229489549303819644288 783165271201909145648566923460 914127372458700660631558817488 925903600113305305488204665213 759591953092186117381932611793 188575272489122793818301194912 463952247371907021798609437027 467669405132000568127145263560 872146844090122495343014654958 561121290219608640344181598136 277016711390098488676975145661
79502884197169399375105820974 482534211706798214808651328230 081284811174502841027019385211 810975665933446128475648233786 034861045432664821339360726024 815209209628292540917153643678 384146951941511609433057270365 310511854807446237996274956735 298336733624406566430860213949 770539217176293176752384674818 082778577134275778960917363717 853710507922796892589235420199 629774771309960518702164201989 1406800700237877659134401712…
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pi
pi and circles ..............................................................................................................
ci r c u
mfe r en ce
s ra diu diameter
� is the symbol used for the number which we call Pi (pronounced ‘pie’). Pi comes from working with circles.
The approximate value of Pi is 3.14159, although the largest calculation for Pi currently stands at ten trillion digits.
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pi
pi and circles .................................................................................................... .......................
......................... Pi is the ratio of a circle’s circumference to its diameter. It is always three .................................................................. times the diameter, plus a ......... little bit. Pi is a constant number, meaning that for all circles of any size, Pi will be the same. Being an irrational number, Pi cannot be expressed exactly as a common fraction. Consequently, its decimal representation never ends and never settles into a permanent repeating pattern. Pi is used in geometry to work out the area of a circle, As well as the volume of a cylinder. .......
However, on a calculator, Pi can only ever be approximate – that is, you cannot show an exact value for Pi with digits on a calculator screen or even on large pieces of paper. This has fascinated mathematicians for a very long time as they continue to find ways to calculate approximations of Pi that are more accurate.
............ Area = �r2 Volume = �r2h
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pi
How I wish I could calculate Pi. ..............
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3.141592.............
piphilology ................................................................................................... The most commonly used versions of Pi are 3.14 and 3.14159. Beyond this, remembering longer versions is a difficult task. To remember them people often use mnemonics and in this case, piphilology (or word-length mnemonics). The number of letters in each word corresponds to a digit. For example, ‘How I wish I could calculate Pi is demonstrated on the opposite page. Longer versions include, ........................................................
....
‘There once was a fellow from Greece, Who forgot Pi’s last decimal piece. So he used electronics To collect Pi mnemonics… Now he’s hooked, and there is no release.’
....
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‘Sir, I bear a rhyme excelling In mystic force, and magic spelling Celestial sprites elucidate All my own striving can’t relate Or locate they who can cogitate And so finally terminate. Finis.’
For many people, it has become a hobby to memorise longer and longer versions of Pi. Whilst the word-length mnenomic method is useful, counting the letters in a word is a slow process, it seems grouping methods are more effective.
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pi
3.14159 The first five are widely known.
26535 Remember the repeated five.
89–79 Remember the repeated nine.
32–38 Remember the repeated three.
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46–26 Remember the repeated six, the four also links this block to the next.
43383 This block has three threes in it.
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27950 This is the start of a four block link. The twos, the ones and the nines.
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28841 ....
...
97169 399–375
A more effective way of memorising Pi is to look for patterns in the sequence of digits, and grouping the digits into blocks. Most people can remember seven things, plus or minus two, which means that one will usually be able to remember between five and nine things at most. So when given a string of numbers to remember, such as 123957001066, it works best to break it down into smaller numbers or blocks of four. ..............
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12 39 57 00 10 66 or 1239 5700 1066
.....
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..........
This technique is most commonly used to memorise telephone numbers, but also enables people to remember hundreds of digits of Pi. An example is shown on the opposite page to remember 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510.
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34
the fibonacci sequence
The fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,...
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The next number is found by adding up the two numbers before it: 2 (1+1), 3 (2+1), 5 (2+3), ...
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Here is a longer list: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, ...
21
............................... When you make squares with those widths, it gives a spiral.
3
5
1 1
2 13
8
the fibonacci numeral systems sequence
fibonacci in the human body ..............................................................................................................
The fibonacci sequence found in the human body, the face and the hands. It is believed that the more accurate the symmetry, the more anatomically correct and beautiful the person is.
The fibonacci sequence, often referred to as the ‘golden ratio’ or the ‘divine proportion’, appears in many biological settings. The human physical form is a good example of this as it highlights many examples of fibonacci numbers, including one, two, three and five. For example, humans have one head and two ears as well as two eyes and one mouth. They have one nose and two nostrils, two arms with three sections – upper arm, lower arm and hand. The two hands have five fingers on each. Beyond this, many believe the fibonacci sequence can be found in the structure of the human body. more specifically. Each part of the index finger, beginning from the tip down to the wrist, is larger than the preceding section by about the ratio of 1.618. It is possibly coincidence, but some say that the proportions of our face, hand, arm, feet,
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1 : 1.618 .............................................................................................
etc. all relate to the sequence.This has led to the belief that the ‘golden section’ marks an ideal shape for the human face. Human DNA molecules are also said to follow the Fibonacci sequence. This means that human DNA has symmetry and to some extent a predictability in how it sequences. Many famous artists have used the fibonacci rectangles in the structure of their artwork. The following page explores the work of Le Corbusier.
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the fibonacci sequence
fibonacci in the human body ................................................................................................... ..........
Left: The Modular Series related to Human Stature, Le Corbusier, c. 1954. Above: The Vitrivian Man. Leonardo da Vinci, c.1490. The drawing is based on the correlations of ideal human proportions in terms of geometry.
The Modulor Man was designed to provide ‘a harmonic measure to the human scale…’
The Modular System It is commonly believed that Leonardo da Vinci’s ‘Vitruvian man’ is an example of the fibonacci sequence in the human anatomy. lthough the ‘Vitruvian Man’ is often shown in connection with the golden ratio, the proportions of the figure do not actually match it, and the text only mentions whole number ratios. However, Le Corbusier explicitly used the golden ratio in his ‘Modulor system’ for the scale of architectural proportion. He saw this system as a continuation of Leonardo da Vinci’s ‘Vitruvian Man’, the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture. He took Leonardo’s suggestion of the golden ratio in human proportions to an extreme.
The Modulor is the proportioning system developed by Le Corbusier, based on the fibonacci series. He believed these proportions to be evident in the human body. The purpose of the Modulor was to ‘maintain the human scale everywhere’. Le Corbusier suggested that the Modulor system would give harmonious proportions to everything, from the sizes of cabinets and door handles to buildings and urban spaces. In a world with an increasing need for mass production, the Modulor was supposed to provide the model for standardization.
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the fibonacci sequence
fibonacci in nature .................................................................................................... The fibonacci numbers are nature’s numbering system. They appear everywhere in nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind.
The diagram on the right shows the fibonacci pattern in a sunflower ...................................... head. This is an example of a double spiral, where the pattern extends to the left and the right.
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the fibonacci sequence
fibonacci in nature .............................................................................................................. The examples here show the spirals that form in plants and shells. In many cases there is a double spiral, one extending from to the left and one to the right. Why do so many natural patterns reflect the fibonacci sequence? Scientists have pondered the question for centuries. In some cases, the correlation may just be coincidence. However, in other situations, the ratio exists because that particular growth pattern evolved as the most effective. In plants, this may mean maximum exposure for light-hungry leaves or maximum seed arrangement.
The sunflower is the most well known example of the fibonacci sequence in nature. The seed head mirrors the double spiral shown on the previous page.
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numeral numerology systems
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Numerology is the study of the purported divine, mystical or othe special relationship between a number and some coinciding observed events.
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Numerology was first recorded c. 1907 but carries a long history of numerogical ideas, traditions and avenues of belief. Today, numerology is often associated with the paranormal, alongside astrology and similar divinatory arts. This chapter will cover the following core elements of Numerology; • • • •
Core numbers Number definitions Master numbers Sun numbers
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introduction ...................................................................................................
There are five building blocks or ‘core numbers’ that make up the Numerology of you: the Life Path number, Expression number, Personality number, Heart’s Desire number and Birth Day number. Each one is derived from your personal birth name and/or birth date and it is believed that the core numbers have a deep-rooted impact on each person. This page breaks down the meaning of the core numbers and gives examples of how to calculate each number.
The expression number is concerned with a person’s natural strengths and weaknesses given at the time of birth. It is often referred to as the ‘destiny number’, as it aims for goals in life. The expression number is the foundation on which a person’s life path can flourish.
expression number Derived from numbers corresponding to letters of the name.* ...................................................................................................
The Life Path number is the single most important number in a person’s numerology chart, because it acts as a blueprint outlining how their life will play out. Opportunities and challenges faced along the way are heavily influenced by the life path number, as is general character and course of action. It encaptures the entire being and will never change, and for those reasons, it is seen to be very special.
what are core numbers? life path number Derived from reducing your full birth date. ................................................ ...................................................................................................
core numbers ......................................................................................................................................................
The birthday number is just that – the day on which a person was born – and it reveals a certain talent that will eventually find its place on their life path. While it is the most insignificant of the core numbers, the birth day number puts a timestamp on who the person is today according to one single aspect that will ultimately impact your life. It is believed that just as individuals were destined to be born on a specific day, they have been given a certain special gift as a result.
birthday number The exact date alone on which a person was born. ...................................................................................................
The reasons behind a person’s actions and their true desires are aspects which stem from the heart’s desires number. It signifies the reasons behind the choices a person makes in all aspects of their life, from career choices to relationships. It is the passion or ‘the burning fire’ within. The soft sounds of the vowels of a person’s name shed light on the inner workings of more subconscious desires.
heart’s desire number Derived from the vowels in the name. ...................................................................................................
The personality number is the first impression people get of a person. It represents the parts of one’s self that they are most ready and willing to reveal, and helps determine just how much is revealed, and to whom. This number acts as a buffer, screening out some people and situations individuals do not want to deal with while welcoming the things in life that relate to their inner nature.
personality number Derived from the consonants in the name. ...................................................................................................
numerology
number definitions .............................................................................................................. The Primal Force Positive: Ones are individualistic and independent, showing leadership and drive. The one is masculine, focused, progressive, an originator and self-starter. They are courageous, strong-willed, rebellious and self-reliant. Negative: Ones can be stubborn, selfish, weak and undisciplined, or a pariah.
The All-Knowing Positive: Twos are sensitive, tactful, diplomatic and cooperative. The twos tend to be peacemakers and are loving, studious and patient. A two may express many musical qualities and also tends to be sensual and intuitive. Negative: Twos are often discontent and can be seen as spoiled. They can be oversensitive.
The creative child Positive: Threes are imaginative, expressive communicators and artists. They are tolerant, joyful, optimistic, inspiring, talented, jovial, youthful and dynamic. Negative: There is a price for how inspirational threes are. They are often vain, extravagant and prone to complaining. They can be intolerant and superficial.
The Salt of the Earth
The Seeker
Positive: Fours are disciplined, strong, stable, pragmatic, down-toearth, dependable, hard-working, extracting, precise, methodical, conscientious, frugal, devoted, patriotic and trustworthy.
Positive: Seven is not just a lucky number. It is also spiritual, intelligent, analytical, focused, introspective, studious, intuitive, knowledgeable, contemplative, persevering, refined, gracious and displays much inner wisdom.
Negative: Fours pay for their stability by tending toward the boring side. This may express itself with a lack of imagination.
Negative: Sevens can be aloof, distant, sarcastic, socially awkward, melancholic and cowardly.
A Dynamic Force
Balance and power
Positive: Fives are energetic, adventurous, and daring people. They also tend to be versatile, flexible, adaptable, curious, social, sensual, witty, quick-thinking, courageous, freedom-loving and worldly people.
Positive: Eights are authoritative, business-minded leaders. They are balanced, materially detached, successful, controlled and realistic. They suit management positions, are efficient, capable, street-smart and good judges of character.
Negative: On the flip side, fives can be unstable, chaotic, selfindulgent, irresponsible or careless.
Negative: The dark side of the eight can be cruel, insensitive, violent, intolerant or greedy.
The caretaker
Global awareness
Positive: Sixes are responsible, loving, self-sacrificing, protective, sympathetic and compassionate. These loyal, maternal figures are domestic, fair and idealistic healers or teachers.
Positive: Nines are helpful, charitable, generous, humanitarian, romantic, cooperative, creative, self-sufficient and proud and self-sacrificing. They are the least judgmental of all the numbers.
Negative: A six can overdo its inherent protectiveness and become anxious, worrisome, suspicious, paranoid and unstable.
Negative: Nines can end up being egocentric, arrogant, self-pitying, sentimental, discontent, fickle, cold or mentally unstable.
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numerology
master numbers ..............................................................................................................
The number eleven represents instinct, and is the most intuitive of all numbers. It is your connection to your subconscious, to gut feeling and knowledge without rationality. Eleven is the dichotomy number, meaning it is both extremely conflicted and also a dynamic catalyst.
The master number twenty-two holds more power than any other number (earning it the nickname 'the master builder'). It is a pragmatic number, a doer, capable of spinning wild dreams into concrete reality. Twentytwo is an ambitious but disciplined number.
The thirty-three is the mover and shaker of the master numbers. With eleven and twenty-two combined in this master number, intuition and dreams reach an entirely new orbit. Thirty-three at full potential means that there is no personal agenda, only a focus on humanitarian issues.
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Septembe r
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Oc to
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ber vem No 23
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26 27
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Ja n ua ry
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r u ar Feb
March
2
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numerology
Ap ril
5
Numerology holds a similar classification of personality traits as zodiac signs. These are called ‘sun numbers’. Based on the day and month a person was born, the sun number is added together and reduced down to a single-digit number. More importantly it describes a person’s personality traits and those of similar sun numbers at a high and simplified level. By looking at the combined numbers of the day and month of birth, numerology deduces your birth date to the first pattern of your existence.
y Ma
6 7 8 9
June
Sun numbers are easy to calculate and never change. Add your month and day of birth, and reduce to a single digit.
11
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.... 23 = 2 + 3 = 5 April = 4
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10
........ 23rd April
.... 5 + 4 = 9
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12
Jul y
23rd April reduces to sun number nine. It is believed that people with a sun number nine are idealists. There is a global consciousness that drives many with this sun number to a career in politics, the military or law enforcement. They can be aloof and distant, but that is a form of self-defense, because they are often not very good with emotional issues, preferring an objective approach.
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lucky and unlucky numbers
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Fear of the number thirteen is a specifically recognised phobia, ‘triskaidekaphobia’.
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Numbers carry different meanings depending on which country you are in. Every culture is characterised by numerical symbolism. Religious beliefs, traditions and cultural legends have been passed down through time to give numbers a perceived lucky or unlucky power. This chapter will explore: • • • •
Numbers across the world Lucky numbers Unlucky numbers Thirteen
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introduction ...................................................................................................
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6
2
7
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9
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lucky and unlucky numbers
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lucky and unlucky numbers
lucky numbers .............................................................................................................. Two is most often considered a good number in Chinese culture following the saying, ‘good things come in pairs’. It is common to repeat characters in product brand names, such as double happiness, which even has its own character. In Cantonese, two is homophone of the characters for ‘easy’ and ‘bright’.
Three is considered a good number in Chinese culture because it sounds like the word ‘alive’. The number sounds similar to the character for ‘birth’ and is also significant since there are three important stages in a man’s life (birth, marriage and death).
Seven is considered lucky in the UK, France, USA and the Netherlands because of its associations in culture. For example, God created the world in seven days, there are seven days of the week, there are seven deadly sins, seven wonders of the world, etc.
Eight is a lucky number in Chinese culture because it sounds like the word meaning to generate wealth. A Hong Kong number plate with the number eight was sold for $640,000. The opening cereomony of thr Summer Olympics in Beijing started at 8 seconds and 8 minutes past 8pm on the 8th August 2008.
Nine is lucky in Norway, it is seen as a sacred number according to Norse mythology. It is considered a good number in Chinese culture because it sounds the same as the word ‘longlasting’. Nine is strongly associated with the Chinese dragon, a symbol of magic and power.
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lucky and unlucky numbers
unlucky numbers .............................................................................................................. Three is unlucky in many cultures, following the phrase that bad luck ‘comes in threes’. In Vietnam there is also a superstition that consider it bad luck to take a photo with three people in it. It is professed that the person in the middle will die soon.
Four is considered unlucky in Korea and Japan because it is pronounced like ‘death’. Following this superstition, many numbered product lines skip the four. For example, there is no series beginning with a four in Nokia cell phones. In East Asia, some buildings do not have a forth floor.
Five is unlucky in China because the number is associated with ‘not’ which clearly indicates ‘not possible’ or ‘not prosperous’, and anything that is negative. However, it can be combined with another unlucky to bring a lucky effect. For example, fifty-four can mean ‘no death’.
Nine is an unlucky number in Japanese culture because in Japanese the word for nine sounds similar to the word for pain, distress or suffering. There are many hospitals that do not have these numbers as the room number or even the floor number.
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lucky and unlucky numbers
Unlucky Number Thirteen
The Last Supper
Fear of the number thirteen is a specifically recognised phobia, ‘triskaidekaphobia’, a word coined in 1911. The superstitious sufferers of triskaidekaphobia try to avoid bad luck by keeping away from anything numbered or labelled thirteen. As a result, companies and manufacturers use another way of numbering or labelling to avoid the number, with hotels and tall buildings being a prime example. It is also considered unlucky to have thirteen guests at a table. Friday the 13th has been considered the unluckiest day of the month. There are a number of theories behind the cause of the association between thirteen and bad luck, but none of them have been accepted as likely.
At Jesus Christ’s last supper, there were thirteen people around the table, counting Christ and the twelve apostles. Some believe this is unlucky because one of those thirteen, Judas was the betrayer of Jesus Christ. Friday the 13th This is considered an unlucky day in Western and Eastern superstition. It is seen as irregular in our calender of twelve months, twelve hours in a day, etc. Apollo 13 The NASA moon mission famous for being a ‘successful failure’ in 1970.
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numbers in language
numbers in language
introduction ..............................................................................................................
Zero tolerance…Nul points…Hung, quarter given…Half inch…Better hal one…One-stop-shop…One-hit won Dutch…Double whammy…Goody t than one…Hat-trick…Ménage à troi lucky…Three score and ten…Three and you are out…Four corners of th column…High-five…At sixes and sev the eight ball…One over the eight… nine…Nine days’ wonder…The who saves nine…Baker’s dozen…Fifteen m seven…A picture is worth a thousan
, drawn and quartered…No lf…Half-hearted…Back to square nder…Two cents worth…Double two-shoes…Two heads are better is…The third degree…Third time e sheets to the wind…Three strikes he earth…A bunch of fives…Fifth vens…Seven-year itch…Behind …Dressed to the nines…On cloud ole nine yards…A stitch in time minutes of fame…Twenty-four nd words. As this book has shown, numbers are everywhere – in history, mathematics, and in culture. This chapter explores
some of the number idioms of the English language and gives a summary of where they originated from.
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numbers in language
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Back to square one
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Two cent’s worth
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Three sheets to the wind
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Four corners of the earth
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A bunch of fives
Meaning: To go back to the beginning, to start again. ‘If this does not work we are back to square one again’.
Origin: ‘Back to square one’ is a classic of folk etymology. There are three widely reported suggestions as to the phrase’s origin: BBC sports commentaries, board games like Snakes and Ladders and playground games like Hopscotch.
.............................................................................................................. Meaning: An individual’s opinion. ‘I thought I would just add my two cent’s worth’
Origin: It has been suggested that ‘two cents’ was the minimum wager required of a new player to enter poker games.
.............................................................................................................. Meaning:Very drunk. ‘He was three sheets to the wind and had long since stopped making sense.’
Origin: This phrase comes from nautical terminology. Sheets are ropes fixed to the lower corners of sails, to hold them in place. If ther are loose then the sails will flap and the boat will lurch about like a drunken sailor.
.............................................................................................................. Meaning: All parts of the earth. ‘She invited relatives from the four corners of the earth to her wedding.’
Origin: From the Bible, the reference to four corners does not imply that the writers of these texts believed that the Earth was flat (although they may well have done). The phrase refers to the four compass points.
.............................................................................................................. Meaning: A fist. The fives are the five fingers. ‘Do you want a bunch of fives?’
Origin: The phrase appears in print in 1825, in Charles Westmacott’s The English Spy, ‘With their bunch of fives.’ It is also reported as appearing slightly earlier, in Boxiana by Pierce Egan, 1821.
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numbers in language
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Six of one, half a dozen of the other
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Sail the Seven seas
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One over the eight
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On cloud nine
Meaning: The two alternatives are equivalent or indifferent. ‘It does not matter which route we take, it is six of one, half a dozen of the other.’
Origin: The origin of this term is unknown, although it is believed to have come from the 1800s.
.............................................................................................................. Meaning: An expression for all the world’s oceans. ‘They sailed the seven seas.’
Origin: The term ‘seven seas’ is mentioned by ancient Hindus, Chinese, and other cultures. The term historically referred to bodies of water along trade routes and regional waters; although in some cases the seas are mythica.
.............................................................................................................. Meaning: The final drink that renders someone drunk. ‘He has had one over the eight’.
Origin: This originated as UK military slang. The first reference to it in print is in Fraser and Gibbons’ Soldier & Sailor Words, 1925, ‘One drink too many, the presumption being that an average man can safely drink eight glasses of beer.’
.............................................................................................................. Meaning: In a state of blissful happiness. ‘If she passes the assessment, she will be on cloud nine.’
Origin: A commonly heard explanation is that the expression originated from one of the classifications of cloud, defined by the US Weather Bureau in the 1950s, in which ‘Cloud Nine’ denotes the fluffy cumulonimbus type that are considered so attractive.
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bibliography
This book has been written as an introduction or summary of numbers. It has covered a broad range of topics concerning numbers, yet this barely scratches the surface of the subject. As mentioned in chapter one, numbers are infinite.It is impossible therefore to gain a complete knowledge. Research was gathered, edited and reworded where necessary, however, none of the factual information was altered or changed in any way. For a deeper understanding of some of the subjects mentioned, see the list of references below for further reading:
Friedman, Erich.
Numbers Around the World
What’s special about this number?
Schimmel, Annamarie.
http://dailyinfographic.com/lucky-unlucky-numbersaround-the-world-infographic.
The Mystery of Numbers
Cook, Theodore.
National Numeracy. http://www.
The Curves of Life.
nationalnumeracy.org.uk/understanding-numbersand-our-number-system/index.html
Basic Mathematics http://www.basicmathematics.com/classification-of-numbers.html Fibonacci Sequence and Golden Section http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/ Fibonacci/fib.html
One Million Digits of Pi http://www.piday.org/million/
Wells, David. The Penguin Dictionary of Curious and Interesting Numbers.
How Many? A Dictionary of Units of Measurement Russ Rowlett and the University of North Carolina at Chapel Hill
Numerology http://www.numerology.com/
Fifty Phrases and Counting http://www.phrases.org.uk/meanings/number-phrases
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