On Indeterminate Forms

“On the Nature of Indeterminate Forms”

Notes compiled from wikipedia By: Jonathan Barlow Gee

“On the Nature of Indeterminate Forms” Notes compiled from wikipedia By: Jonathan Barlow Gee Is hereby public domain and anti-© This, Sunday, December 12, 2021 AD.

“On the Nature of Number Sets” Number Sets Defined Main types Types of integer Algebraic numbers Non-standard numbers Computability and definability Natural numbers Integers Rational numbers Real numbers Complex numbers

“On the Nature of Zero” Zero Defined A Brief History of Zero Some Mathematical Traits of Zero

“On the Nature of One” One Defined A Brief History of One Some Mathematical Traits of One Some Mathematical Traits of Negative One

“On the Nature of Infinity” Infinity Defined A Brief History of Infinity Some Mathematical Traits of Infinity The Lemniscate

“General Notes” On Trans-finites and Number Sets On Indeterminate Forms Some Brief Conclusions

“On the Nature of Number Sets” Introduction to: “Indeterminate Forms”

Notes compiled from wikipedia By: Jonathan Barlow Gee

Number Sets Defined “Numbers can be classified into sets, called number systems. The major categories of numbers are as follows:

There is generally no problem in identifying each number system with a proper subset of the next one (by abuse of notation), because each of these number systems is canonically isomorphic to a proper subset of the next one. The resulting hierarchy allows, for example, to talk, formally correctly, about real numbers that are rational numbers, and is expressed symbolically by writing…

Main types “Natural numbers (N): The counting numbers {1, 2, 3, ...} are commonly called natural numbers; however, other definitions include 0, so that the non-negative integers {0, 1, 2, 3, ...} are also called natural numbers. Natural numbers including 0 are also called whole numbers.

Integers (Z): Positive and negative counting numbers, as well as zero: {..., −3, −2, −1, 0, 1, 2, 3, ...}.

Rational numbers (Q): Numbers that can be expressed as a ratio of an integer to a non-zero integer. All integers are rational, but there are rational numbers that are not integers, such as −2/9.

Real numbers (R): Numbers that can represent a distance along a line. They can be positive, negative, or zero. All rational numbers are real, but the converse is not true.

Irrational numbers: Real numbers that are not rational.

Imaginary numbers: Numbers that equal the product of a real number and the square root of −1.

The number 0 is both real and purely imaginary.

Complex numbers (C): Includes real numbers, imaginary numbers, and sums and differences of real and imaginary numbers.

Hypercomplex numbers include various number-system extensions: quaternions (H), octonions (O), and other less common variants. p-adic numbers: Various number systems constructed using limits of rational numbers, according to notions of "limit" different from the one used to construct the real numbers.

Types of integer Even and odd numbers: An integer is even if it is a multiple of 2, and is odd otherwise.

Prime number: A positive integer with exactly two positive divisors: itself and 1. The primes form an infinite sequence 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...

Composite number: A positive integer that can be factored into a product of smaller positive integers. Every integer greater than one is either prime or composite.

Polygonal numbers: These are numbers that can be represented as dots that are arranged in the shape of a regular polygon, including Triangular numbers, Square numbers, Pentagonal numbers, Hexagonal numbers, Heptagonal numbers, Octagonal numbers, Nonagonal numbers, Decagonal numbers, Hendecagonal numbers, and Dodecagonal numbers.

There are many other famous integer sequences, such as the sequence of Fibonacci numbers, the sequence of factorials, the sequence of perfect numbers, and so forth.

Algebraic numbers Algebraic number: Any number that is the root of a non-zero polynomial with rational coefficients.

Transcendental number: Any real or complex number that is not algebraic. Examples include e and π.

Trigonometric number: Any number that is the sine or cosine of a rational multiple of π.

Quadratic surd: A root of a quadratic equation with rational coefficients. Such a number is algebraic and can be expressed as the sum of a rational number and the square root of a rational number.

Constructible number: A number representing a length that can be constructed using a compass and straightedge. Constructible numbers form a subfield of the field of algebraic numbers, and include the quadratic surds.

Algebraic integer: A root of a monic polynomial with integer coefficients.

Non-standard numbers Transfinite numbers: Numbers that are greater than any natural number.

Ordinal numbers: Finite and infinite numbers used to describe the order type of well-ordered sets.

Cardinal numbers: Finite and infinite numbers used to describe the cardinalities of sets.

Infinitesimals: These are smaller than any positive real number, but are nonetheless greater than zero. These were used in the initial development of calculus, and are used in synthetic differential geometry.

Hyperreal numbers: The numbers used in non-standard analysis. These include infinite and infinitesimal numbers which possess certain properties of the real numbers.

Surreal numbers: A number system that includes the hyperreal numbers as well as the ordinals.

Computability and definability Computable number: A real number whose digits can be computed by some algorithm.

Definable number: A real number that can be defined uniquely using a first-order formula with one free variable in the language of set theory.”

Natural numbers “In mathematics, the natural numbers are those numbers used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In common mathematical terminology, words colloquially used for counting are "cardinal numbers", and words used for ordering are "ordinal numbers". Some definitions begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, ..., whereas others start with 1, corresponding to the positive integers 1, 2, 3, ... Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural numbers are a basis from which many other number sets may be built by extension: the integers, by including (if not yet in) the neutral element 0 and an additive inverse (−n) for each nonzero natural number n; the rational numbers, by including a multiplicative inverse (1/n) for each nonzero integer n (and also the product of these inverses by integers); the real numbers by including with the rationals the limits of (converging) Cauchy sequences of rationals; the complex numbers, by including with the real numbers the unresolved square root of minus one (and also the sums and products thereof); and so on. This chain of extensions make the natural numbers canonically embedded (identified) in the other number systems. Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory.

Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, particularly in primary school education, natural numbers may be called counting numbers to intuitively exclude the negative integers and zero, and also to contrast the discreteness of counting to the continuity of measurement—a hallmark characteristic of real numbers. The set of natural numbers is an infinite set. By definition, this kind of infinity is called countable infinity. All sets that can be put into a bijective relation to the natural numbers are said to have this kind of infinity. This is also expressed by saying that the cardinal number of the set is aleph-nought (ℵ0).”

Integers “An integer (from the Latin integer meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and √2 are not. The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, ...), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., −1, −2, −3, ...). The set of integers is often denoted by the letter "Z"— standing originally for the German word Zahlen ("numbers"). Z is a subset of the set of all rational numbers Q, which in turn is a subset of the real numbers R. Like the natural numbers, Z is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.”

Rational numbers “In mathematics, a rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. The decimal expansion of a rational number either terminates after a finite number of digits (example: 3/4 = 0.75), or eventually begins to repeat the same finite sequence of digits over and over (example: 9/44 = 0.20454545...). Conversely, any repeating or terminating decimal represents a rational number. A real number that is not rational is called irrational. Irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers together with addition and multiplication form a field which contains the integers, and is contained in any field containing the integers.”

Real numbers “In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line (or alternatively, a quantity that can be represented as an infinite decimal expansion).

The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2 (1.41421356..., the square root of 2, an irrational algebraic number). Included within the irrationals are the real transcendental numbers, such as π (3.14159265…). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more. Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced.

Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, where each consecutive digit is measured in units one-tenth the size of the previous one.

The real line can be thought of as a part of the complex plane, and the real numbers can be thought of as a part of the complex numbers. The set of all real numbers is uncountable, in the sense that while both the set of all natural numbers and the set of all real numbers are infinite sets, there can be no one-to-one function from the real numbers to the natural numbers. In fact, the cardinality of the set of all real numbers, denoted by c and called the cardinality of the continuum, is strictly greater than the cardinality of the set of all natural numbers (denoted ℵ0 or ‘aleph-naught'). The real numbers make up an infinite set of numbers that cannot be injectively mapped to the infinite set of natural numbers, i.e., there are uncountably infinitely many real numbers, whereas the natural numbers are called countably infinite. This establishes that in some sense, there are more real numbers than there are elements in any countable set. There is a hierarchy of countably infinite subsets of the real numbers, e.g., the integers, the rationals, the algebraic numbers and the computable numbers, each set being a proper subset of the next in the sequence. The complements of all these sets (irrational, transcendental, and non-computable real numbers) in the reals are all uncountably infinite sets.”

Complex numbers “In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted i, called the imaginary unit, and satisfying the equation i^2 = −1. Moreover, every complex number can be expressed in the form a + bi, where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number a + bi, a is called the real part and b is called the imaginary part. Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule i^2 = −1 combined with the associative, commutative and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field that has the real numbers as a subfield. The complex numbers form also a real vector space of dimension two, with {1, i} as a standard basis. This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the real line which is identified to the horizontal axis of the complex plane. The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with respect to the real axis. The complex absolute value is a Euclidean norm. In summary, the complex numbers form a rich structure that is simultaneously an algebraically closed field, a commutative algebra over the reals, and a Euclidean vector space of dimension two.”

https://en.wikipedia.org/wiki/Number https://en.wikipedia.org/wiki/List_of_types_of_numbers https://en.wikipedia.org/wiki/Natural_number https://en.wikipedia.org/wiki/Integer https://en.wikipedia.org/wiki/Rational_number https://en.wikipedia.org/wiki/Real_number https://en.wikipedia.org/wiki/Complex_number

“On the Nature of Zero” “Indeterminate Forms” (Part 1)

Notes compiled from wikipedia By: Jonathan Barlow Gee

Zero Defined “0 (zero) is a number, and the numerical digit used to represent that number in numerals.” A “number” is defined as “a mathematical object used to count, measure, and label” and a “numerical digit” is defined as “a single symbol used alone (such as "2") or in combinations (such as "25"), to represent numbers in a positional numeral system. …

More universally, individual numbers can be represented by symbols, called numerals. … As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a numeral system, which is an organized way to represent any number. The most common numeral system is the Hindu–Arabic numeral system, which allows for the representation of any number using a combination of ten fundamental numeric symbols, called digits. … Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation.”

“The use of 0 as a number should be distinguished from its use as a placeholder numeral in place-value systems. As a value or a number, zero is not the same as the digit zero, used in numeral systems with positional notation. Successive positions of digits have higher weights, so the digit zero is used inside a numeral to skip a position and give appropriate weights to the preceding and following digits.”

A Brief History of Zero “Many ancient texts used 0. Babylonian and Egyptian texts used it. The first known system with place value was the Mesopotamian base-60 system (c. 3400 BC) and the earliest known base-10 system dates to 3100 BC in Egypt. By the middle of the 2nd millennium BC, Babylonian mathematics had a sophisticated sexagesimal positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. Ancient Egyptian numerals were of base-10. They used hieroglyphs for the digits and were not positional. By 1770 BC, the Egyptians had a symbol for zero in accounting texts. Egyptians used the word nfr to denote a zero balance in double entry accounting. The symbol nfr - meaning beautiful, was also used to indicate the base level in drawings of tombs and pyramids, and distances were measured relative to the base line as being above or below this.”

“The late Olmec people of south-central Mexico began to use a symbol for zero, a shell glyph, in the New World - possibly by the 4th century BC but certainly by 40 BC - which became an integral part of Mayan numerals and the Mayan calendar. … Since the eight earliest Long Count dates appear outside the Maya homeland, it is generally believed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs. Many of the earliest Long Count dates were found within the Olmec heartland, although the Olmec civilization ended by the 4th century BC, several centuries before the earliest known Long Count dates. Although zero became an integral part of Maya numerals, with a different, empty tortoise-like "shell shape" - used for many depictions of the "zero" numeral, it is assumed not to have influenced Old World numeral systems.”

“The ancient Greeks had no symbol for zero (μηδέν), and did not use a digit placeholder for it. The paradoxes of Zeno of Elea (c. 490–430 BC) depend in large part on this uncertain interpretation of zero. By AD 150, Claudius Ptolemy (c. 100 - c. 170 AD), influenced by Hipparchus (c. 190 - c. 120 BC) and the Babylonians, was using a symbol for zero in his work on mathematical astronomy called the Syntaxis Mathematica, also known as the Almagest. This Hellenistic zero was perhaps the earliest documented use of a numeral representing zero in the Old World. Ptolemy used it many times in his Almagest (VI.8) for the magnitude of solar and lunar eclipses. It represented the value of both digits and minutes of immersion at first and last contact. Minutes of immersion was tabulated from 0′0″ to 31′20″ to 0′0″, where 0′0″ used the symbol as a placeholder in two positions of his sexagesimal positional numeral system, while the combination meant a zero angle. … Ptolemy's symbol was a placeholder as well as a number used by two continuous mathematical functions, one within another, so it meant zero, not none. The earliest use of zero in the calculation of Julian Easter occurred before AD 311, at the first entry in a table of epacts as preserved in an Ethiopic document for the years AD 311 to 369, using a Ge'ez word for "none" (English translation is "0" elsewhere) alongside Ge'ez numerals (based on Greek numerals), which was translated from an equivalent table published by the Church of Alexandria in Medieval Greek. This use was repeated in AD 525 in an equivalent table, that was translated via the Latin nulla or "none" by Dionysius Exiguus, alongside Roman numerals. When division produced zero as a remainder, nihil, meaning "nothing", was used. These medieval zeros were used by all future medieval calculators of Easter.”

“Indian texts used a Sanskrit word Shunye or shunya to refer to the concept of void. In mathematics texts this word often refers to the number zero. … The concept of zero as a written digit in the decimal place value notation was developed in India, presumably as early as during the Gupta period (c. 5th century), with the oldest unambiguous evidence dating to the 7th century. The Lokavibhāga, a Jain text on cosmology surviving in a medieval Sanskrit translation of the Prakrit original, which is internally dated to AD 458 (Saka era 380), uses a decimal place-value system, including a zero. In this text, śūnya ("void, empty") is also used to refer to zero. … Rules governing the use of zero appeared in Brahmagupta's Brahmasputha Siddhanta (7th century), which states the sum of zero with itself as zero, and (incorrectly) division by zero as: ““A positive or negative number when divided by zero is a fraction with the zero as denominator.

Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.””

“The Brāhmasphuṭasiddhānta ("Correctly Established Doctrine of Brahma") is the main work of Brahmagupta, written c. 628 AD. This text of mathematical astronomy contains significant mathematical content, including a good understanding of the role of zero, rules for manipulating both negative and positive numbers, a method for computing square roots, methods of solving linear and quadratic equations, and rules for summing series, Brahmagupta's identity, and Brahmagupta’s theorem. Brāhmasphuṭasiddhānta is one of the first books to provide concrete ideas on positive numbers, negative numbers, and zero:

““The sum of two positive quantities is positive The sum of two negative quantities is negative The sum of zero and a negative number is negative The sum of zero and a positive number is positive The sum of zero and zero is zero The sum of a positive and a negative is their difference; or, if they are equal, zero In subtraction, the less is to be taken from the greater, positive from positive In subtraction, the less is to be taken from the greater, negative from negative When the greater however, is subtracted from the less, the difference is reversed When positive is to be subtracted from negative, and negative from positive, they must be added together The product of a negative quantity and a positive quantity is negative The product of two negative quantities is positive The product of two positive quantities is positive Positive divided by positive or negative by negative is positive Positive divided by negative is negative. Negative divided by positive is negative Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator A positive or negative number when divided by zero is a fraction with the zero as denominator Zero divided by zero is zero””

“The last two of these rules are notable as the earliest attempt to define division by zero, even though they are not compatible with modern number theory (division by zero is undefined for a field). … In 830, Mahāvīra unsuccessfully tried to correct Brahmagupta's mistake in his book in Ganita Sara Samgraha: ““A number remains unchanged when divided by zero.””

“The Arabic-language inheritance of science was largely Greek, followed by Hindu influences. In 773, at Al-Mansur's behest, translations were made of many ancient treatises including Greek, Roman, Indian, and others. In AD 813, astronomical tables were prepared by a Persian mathematician, Muḥammad ibn Mūsā al-Khwārizmī (c. 780 – c. 850), using Hindu numerals; and about 825, he published a book synthesizing Greek and Hindu knowledge that also contained his own contribution to mathematics including an explanation on the use of zero. This book was later translated into Latin in the 12th century under the title Algoritmi de numero Indorum. This title means "al-Khwarizmi on the Numerals of the Indians". The word "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name, and the word "Algorithm" or "Algorism" started to acquire the meaning of any arithmetic based on decimals. Muhammad ibn Ahmad al-Khwarizmi, in 976, stated that if no number appears in the place of tens in a calculation, a little circle should be used "to keep the rows". This circle was called ṣifr.”

“The Hindu–Arabic numeral system (base 10) reached Western Europe in the 11th century, via Al-Andalus, through Spanish Muslims, the Moors, together with knowledge of classical astronomy and instruments like the astrolabe; Gerbert of Aurillac (c. 946 – 12 May 1003) is credited with reintroducing the lost teachings into Catholic Europe. For this reason, the numerals came to be known in Europe as "Arabic numerals". The Italian mathematician Fibonacci or Leonardo of Pisa (c. 1170 – c. 1240–50) was instrumental in bringing the system into European mathematics in 1202. Fibonnaci used the phrase "sign 0", indicating zero is a sign with which to do operations like addition or multiplication.

From the 13th century, manuals on calculation (adding, multiplying, extracting roots, etc.) became common in Europe where they were called algorismus after the Persian mathematician al-Khwārizmī. The most popular was written by Johannes de Sacrobosco (about 1235) and was one of the earliest scientific books to be printed (in 1488). Until the late 15th century, Hindu–Arabic numerals seem to have predominated among mathematicians, while merchants preferred to use Roman numerals.”

Some Mathematical Traits of Zero “0 is the integer immediately preceding 1. Zero is an even number because it is divisible by 2 with no remainder. 0 is neither positive nor negative, or both positive and negative.

Many definitions include 0 as a natural number, in which case it is the only natural number that is not positive. Zero is a number which quantifies a count or an amount of null size. The number 0 is the smallest non-negative integer. The natural number following 0 is 1 and no natural number precedes 0. The number 0 may or may not be considered a natural number, but it is an integer, and hence a rational number and a real number (as well as an algebraic number and a complex number). The number 0 is neither positive nor negative, and is usually displayed as the central number in a number line. It is neither a prime number nor a composite number. It cannot be prime because it has an infinite number of factors, and cannot be composite because it cannot be expressed as a product of prime numbers (as 0 must always be one of the factors).

Zero is, however, even (i.e. a multiple of 2, as well as being a multiple of any other integer, rational, or real number). The following are some basic (elementary) rules for dealing with the number 0. These rules apply for any real or complex number x, unless otherwise stated: ““Addition: x + 0 = 0 + x = x.

That is, 0 is an identity element (or neutral element) with respect to addition.

Subtraction: x − 0 = x and 0 − x = −x.

Multiplication: x · 0 = 0 · x = 0.

Division: 0/x = 0, for nonzero x.

But x/0 is undefined, because 0 has no multiplicative inverse (no real number multiplied by 0 produces 1), a consequence of the previous rule.

Exponentiation: x^0 = x/x = 1, except that the case x = 0 may be left undefined in some contexts.

For all positive real x, 0^x = 0.””

“The sum of 0 numbers (the empty sum) is 0, and the product of 0 numbers (the empty product) is 1. The factorial 0! evaluates to 1, as a special case of the empty product.”

“In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. A graphical illustration of a nildimensional space is a point. … The zero-dimensional hypersphere is a pair of points. The zero-dimensional ball is a point.”

“In mathematics, division by zero is division where the divisor (denominator) is zero.

Such a division can be formally expressed as a/0 where a is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is no number which, when multiplied by 0, gives a (assuming a ≠ 0), and so division by zero is undefined. Since any number multiplied by zero is zero, the expression 0/0 is also undefined; when it is the form of a limit, it is an indeterminate form. There are mathematical structures in which a/0 is defined for some a such as in the Riemann sphere (a model of the extended complex plane) and the Projectively extended real line; however, such structures do not satisfy every ordinary rule of arithmetic (the field axioms). … In general, a single value can’t be assigned to a fraction where the denominator is 0 so the value remains undefined. … A formal calculation is one carried out using rules of arithmetic, without consideration of whether the result of the calculation is well-defined. Thus, it is sometimes useful to think of a/0, where a ≠ 0, as being ∞. This infinity can be either positive, negative, or unsigned, depending on context. Although division by zero cannot be sensibly defined with real numbers and integers, it is possible to consistently define it, or similar operations, in other mathematical structures. In the hyperreal numbers and the surreal numbers, division by zero is still impossible, but division by non-zero infinitesimals is possible. In distribution theory one can extend the function 1/x to a distribution on the whole space of real numbers. In matrix algebra (or linear algebra in general), one can define a pseudo-division, by setting a/b = ab+, in which b+ represents the pseudoinverse of b. It can be proven that if b−1 exists, then b+ = b−1. If b equals 0, then b+ = 0. Any number system that forms a commutative ring—for instance, the integers, the real numbers, and the complex numbers—can be extended to a wheel. In field theory, the expression a/b is only shorthand for the formal expression ab^−1, where b^−1 is the multiplicative inverse of b. Since the field axioms only guarantee the existence of such inverses for nonzero elements, this expression has no meaning when b is zero. Modern texts, that define fields as a special type of ring, include the axiom 0 ≠ 1 for fields (or its equivalent) so that the zero ring is excluded from being a field. In the zero ring, division by zero is possible, which shows that the other field axioms are not sufficient to exclude division by zero in a field.”

“In abstract algebra, an element a of a ring R is called a left zero divisor if there exists a nonzero x in R such that ax = 0, or equivalently if the map from R to R that sends x to ax is not injective. Similarly, an element a of a ring is called a right zero divisor if there exists a nonzero y in R such that ya = 0. This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor. An element a that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero x such that ax = 0 may be different from the nonzero y such that ya = 0). If the ring is commutative, then the left and right zero divisors are the same. An element of a ring that is not a left zero divisor is called left regular or left cancellable. Similarly, an

element of a ring that is not a right zero divisor is called right regular or right cancellable. An element of a ring that is left and right cancellable, and is hence not a zero divisor, is called regular or cancellable, or a non-zero-divisor. A zero divisor that is nonzero is called a nonzero zero divisor or a nontrivial zero divisor. A nonzero ring with no nontrivial zero divisors is called a domain. … The only zero divisor of the ring Z of integers is 0. The ring of integers modulo a prime number has no zero divisors other than 0. Since every nonzero element is a unit, this ring is a finite field. More generally, a division ring has no zero divisors except 0. A nonzero commutative ring whose only zero divisor is 0 is called an integral domain. In the ring of n-by-n matrices over a field, the left and right zero divisors coincide; they are precisely the singular matrices. In the ring of n-by-n matrices over an integral domain, the zero divisors are precisely the matrices with determinant zero. If R is a ring other than the zero ring, then 0 is a (two-sided) zero divisor, because any nonzero element x satisfies 0x = 0 = x0. If R is the zero ring, in which 0 = 1, then 0 is not a zero divisor, because there is no nonzero element that when multiplied by 0 yields 0.”

“In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, it is the following assertion: ““If ab = 0, then a = 0 or b = 0.””

“In general, a ring which satisfies the zero-product property is called a domain. A commutative domain with a multiplicative identity element is called an integral domain.

Any field is an integral domain; in fact, any subring of a field is an integral domain (as long as it contains 1). Similarly, any subring of a skew field is a domain. Thus, the zeroproduct property holds for any subring of a skew field. If p is a prime number, then the ring of integers modulo p has the zero-product property (in fact, it is a field). The Gaussian integers are an integral domain because they are a subring of the complex numbers. In the strictly skew field of quaternions, the zero-product property holds. This ring is not an integral domain, because the multiplication is not commutative. The set of nonnegative integers {0,1,2,…} is not a ring (being instead a semiring), but it does satisfy the zero-product property. Analytic functions have the zero-product property.”

“In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this substitution does not provide sufficient information to determine the original limit, then the expression is called an indeterminate form. More specifically, an indeterminate form is a mathematical expression involving at most two of 0, 1 or ∞, obtained by applying the algebraic limit theorem in the process of attempting to determine a limit, which fails to restrict that limit to one specific value or infinity, and thus does not determine the limit being sought. A limit confirmed to be infinity is not indeterminate since it has been determined to have a specific value (infinity). There are seven indeterminate forms which are typically considered in the literature: ““0/0, ∞/∞, 0 ∙∞, ∞-∞, 0^0, 1^∞, and ∞^0.””

“The most common example of an indeterminate form occurs when determining the limit of the ratio of two functions, in which both of these functions tend to zero in the limit, and is referred to as "the indeterminate form 0/0.” For example, as x approaches 0, the ratios x/x^3, x/x, and x^2/x go to ∞, 1, and 0 respectively. In each case, if the limits of the numerator and denominator are substituted, the resulting expression is 0/0, which is undefined. In a loose manner of speaking, 0/0 can take on the values 0, 1, or ∞, and it is easy to construct similar examples for which the limit is any particular value.”

“Zero to the power of zero, denoted by 0^0, is a mathematical expression with no agreedupon value. The most common possibilities are 1 or leaving the expression undefined, with justifications existing for each, depending on context. In algebra and combinatorics, the generally agreed upon value is 0^0 = 1, whereas in mathematical analysis, the expression is sometimes left undefined. Many widely used formulas involving naturalnumber exponents require 0^0 to be defined as 1. For example, the following three interpretations of b^0 make just as much sense for b = 0 as they do for positive integers b: The interpretation of b^0 as an empty product assigns it the value 1. The combinatorial interpretation of b^0 is the number of 0-tuples of elements from a belement set; there is exactly one 0-tuple. The set-theoretic interpretation of b^0 is the number of functions from the empty set to a b-element set; there is exactly one such function, namely, the empty function. All three of these specialize to give 0^0 = 1.” https://en.wikipedia.org/wiki/Number https://en.wikipedia.org/wiki/0 https://en.wikipedia.org/wiki/Brāhmasphuṭasiddhānta https://en.wikipedia.org/wiki/Zero-dimensional_space https://en.wikipedia.org/wiki/Division_by_zero https://en.wikipedia.org/wiki/Zero-product_property https://en.wikipedia.org/wiki/Indeterminate_form https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero

“On the Nature of One” “Indeterminate Forms” (Part 2)

Notes compiled from wikipedia By: Jonathan Barlow Gee

One Defined “1 (one, also called unit, and unity) is a number and a numerical digit used to represent that number in numerals. It represents a single entity, the unit of counting or measurement. For example, a line segment of unit length is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0. It commonly denotes the first, leading or top thing in a group. The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 equals the same number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention not considered a prime number; although universally accepted today, this fact was controversial until the mid-20th century. Any number multiplied by one remains that number, as one is the identity for multiplication. As a result, 1 is its own factorial, its own square and square root, its own cube and cube root, and so on. One is also the result of the empty product, as any number multiplied by one is itself. It is also the only natural number that is neither composite nor prime with respect to division, but is instead considered a unit (meaning of ring theory).”

A Brief History of One “The word one can be used as a noun, an adjective, and a pronoun. It comes from the English word an, which comes from the Proto-Germanic root *ainaz. The Proto-Germanic root *ainaz comes from the Proto-Indo-European root *oi-no-.

Compare the Proto-Germanic root *ainaz to Old Frisian an, Gothic ains, Danish en, Dutch een, German eins and Old Norse einn.

Compare the Proto-Indo-European root *oi-no- (which means "one, single") to Greek oinos (which means "ace" on dice), Latin unus (one), Old Persian aivam, Old Church Slavonic -inu and ino-, Lithuanian vienas, Old Irish oin and Breton un (one).”

Some Mathematical Traits of One “in arithmetic (algebra) and calculus, the natural number that follows 0 and the multiplicative identity element of the integers, real numbers and complex numbers; more generally, in algebra, the multiplicative identity (also called unity), usually of a group or a ring.

Formalizations of the natural numbers have their own representations of 1. In the Peano axioms, 1 is the successor of 0. In Principia Mathematica, it is defined as the set of all singletons (sets with one element), and in the Von Neumann cardinal assignment of natural numbers, it is defined as the set {0}.

In a multiplicative group or monoid, the identity element is sometimes denoted 1, but e (from the German Einheit, "unity") is also traditional. However, 1 is especially common for the multiplicative identity of a ring, i.e., when an addition and 0 are also present.

When such a ring has characteristic n not equal to 0, the element called 1 has the property that n1 = 1n = 0 (where this 0 is the additive identity of the ring). Important examples are finite fields.

By definition, 1 is the magnitude, absolute value, or norm of a unit complex number, unit vector, and a unit matrix (more usually called an identity matrix). Note that the term unit matrix is sometimes used to mean something quite different.

By definition, 1 is the probability of an event that is absolutely or almost certain to occur.

In category theory, 1 is sometimes used to denote the terminal object of a category.

In number theory, 1 is the value of Legendre's constant, which was introduced in 1808

by Adrien-Marie Legendre in expressing the asymptotic behavior of the prime-counting function. Legendre's constant was originally conjectured to be approximately 1.08366, but was proven to equal exactly 1 in 1899.

Since the base 1 exponential function (1x) always equals 1, its inverse does not exist (which would be called the logarithm base 1 if it did exist).

There are two ways to write the real number 1 as a recurring decimal: as 1.000..., and as 0.999.... 1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few.

In many mathematical and engineering problems, numeric values are typically normalized to fall within the unit interval from 0 to 1, where 1 usually represents the maximum possible value in the range of parameters. Likewise, vectors are often normalized into unit vectors (i.e., vectors of magnitude one), because these often have more desirable properties. Functions, too, are often normalized by the condition that they have integral one, maximum value one, or square integral one, depending on the application.

Because of the multiplicative identity, if f(x) is a multiplicative function, then f(1) must be equal to 1.

It is also the first and second number in the Fibonacci sequence (0 being the zeroth) and is the first number in many other mathematical sequences.

The definition of a field requires that 1 must not be equal to 0. Thus, there are no fields of characteristic 1. Nevertheless, abstract algebra can consider the field with one element, which is not a singleton and is not a set at all.

1 is by convention neither a prime number nor a composite number, but a unit (meaning of ring theory) like −1 and, in the Gaussian integers, i and −i.

The fundamental theorem of arithmetic guarantees unique factorization over the integers only up to units. For example, 4 = 22, but if units are included, is also equal to, say, (−1)6 × 123 × 22, among infinitely many similar "factorizations".

1 appears to meet the naïve definition of a prime number, being evenly divisible only by 1 and itself (also 1). As such, some mathematicians considered it a prime number as late as the middle of the 20th century, but mathematical consensus has generally and since then universally been to exclude it for a variety of reasons (such as complicating the fundamental theorem of arithmetic and other theorems related to prime numbers).

1 is the only positive integer divisible by exactly one positive integer, whereas prime numbers are divisible by exactly two positive integers, composite numbers are divisible by more than two positive integers, and zero is divisible by all positive integers.”

Some Mathematical Traits of Negative One “In mathematics, −1 (also known as negative one or minus one) is the additive inverse of 1, that is, the number that when added to 1 gives the additive identity element, 0. It is the negative integer greater than negative two (−2) and less than 0.

Multiplying a number by −1 is equivalent to changing the sign of the number – that is, for any x we have (−1) ⋅ x = −x. This can be proved using the distributive law and the axiom that 1 is the multiplicative identity: x + (−1) ⋅ x = 1 ⋅ x + (−1) ⋅ x = (1 + (−1)) ⋅ x = 0 ⋅ x = 0.

Here we have used the fact that any number x times 0 equals 0, which follows by cancellation from the equation 0 ⋅ x = (0 + 0) ⋅ x = 0 ⋅ x + 0 ⋅ x.

In other words, x + (−1) ⋅ x = 0, so (−1) ⋅ x is the additive inverse of x, i.e. (−1) ⋅ x = −x.

The square of −1, i.e. −1 multiplied by −1, equals 1. As a consequence, a product of two negative numbers is positive.

For an algebraic proof of this result, start with the equation 0 = −1 ⋅ 0 = −1 ⋅ [1 + (−1)].

The first equality follows from the above result, and the second follows from the definition of −1 as additive inverse of 1: it is precisely that number which when added to 1 gives 0. Now, using the distributive law, we see that 0 = −1 ⋅ [1 + (−1)] = −1 ⋅ 1 + (−1) ⋅ (−1) = −1 + (−1) ⋅ (−1).

The third equality follows from the fact that 1 is a multiplicative identity. But now adding 1 to both sides of this last equation implies (−1) ⋅ (−1) = 1.

The above arguments hold in any ring, a concept of abstract algebra generalizing integers and real numbers.

Although there are no real square roots of −1, the complex number i satisfies i^2 = −1, and as such can be considered as a square root of −1. The only other complex number whose square is −1 is −i because there are exactly two square roots of any non-zero complex number, which follows from the fundamental theorem of algebra. In the algebra of quaternions – where the fundamental theorem does not apply – which contains the complex numbers, the equation x^2 = −1 has infinitely many solutions.

Exponentiation of a non-zero real number can be extended to negative integers. We make the definition that x^−1 = 1/x, meaning that we define raising a number to the power −1 to have the same effect as taking its reciprocal. This definition is then extended to negative integers, preserving the exponential law x^a ⋅ x^b = x^(a + b) for real numbers a and b.

Exponentiation to negative integers can be extended to invertible elements of a ring, by defining x^−1 as the multiplicative inverse of x.

A −1 that appears as a superscript of a function does not mean taking the (pointwise) reciprocal of that function, but rather the inverse function of the function. For example, sin^−1(x) is a notation for the arcsine function, and in general f^−1(x) denotes the inverse function of f(x),. When a subset of the codomain is specified inside the function, it instead denotes the preimage of that subset under the function.

-1 bears relation to Euler's identity since e^iπ = −1.”

https://en.wikipedia.org/wiki/1 https://en.wikipedia.org/wiki/%E2%88%921

“On the Nature of Infinity” “Indeterminate Forms” (Part 3)

Notes compiled from wikipedia By: Jonathan Barlow Gee

Infinity Defined “Infinity is a concept referring to that which is boundless, endless, or larger than any number. It is often denoted by the infinity symbol (∞). Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the ‘infinitesimal’ calculus (the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations), mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object. The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets.”

A Brief History of Infinity “Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept. The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite". Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion. … Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him. ““In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.””

- as recounted by Aristotle, Physics VI:9, 239b15 The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:

Enumerable: lowest, intermediate, and highest Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable Infinite: nearly infinite, truly infinite, infinitely infinite

In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation ∞ for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of 1/∞. But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c." In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work ‘De analysi per aequationes numero terminorum infinitas.’ The exponential series, i.e. tending toward infinity, was discovered by Newton and is contained within the Analysis. The treatise contains also the sine series and cosine series and arc series, the logarithmic series and the binomial series.”

Some Mathematical Traits of Infinity “Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.

In real analysis, the symbol ∞ called "infinity", is used to denote an unbounded limit.

The notation x → ∞ means that x increases without bound, and x → -∞ means that x decreases without bound. Infinity can also be used to describe infinite series. In addition to defining a limit, infinity can also be used as a value in the extended real number system. Points labeled +∞ and -∞ can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat +∞ and -∞ as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a ‘line at infinity’ (in geometry and topology, the line at infinity is a projective line that is added to the real ‘affine’ plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane) in plane geometry, a ‘plane at infinity’ (in projective geometry, a plane at infinity is the hyperplane at infinity of a three dimensional projective space or to any plane contained in the hyperplane at infinity of any projective space of higher dimension) in three-dimensional space, and a ‘hyperplane at infinity’ for general dimensions, each consisting of ‘points at infinity’ (in geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line).

In complex analysis the symbol ∞, called "infinity", denotes an unsigned infinite limit. x → ∞ means that the magnitude |x| of x grows beyond any assigned value. A point labeled ∞ can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a onedimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the

extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely z/0 = ∞ for any nonzero complex number z. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of ∞ at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations.

The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets. Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers.

Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences.

Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable.

One of Cantor's most important results was that the cardinality of the continuum c is greater than that of the natural numbers ℵ0; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that c = 2^ℵ0 > ℵ0. The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, c = ℵ1 = ℶ1. The cardinal ℶ0 = ℵ0 is the cardinality of any countably infinite set such as the set N of natural numbers, so that ℶ0 = |N|.

In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of

infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets. The term transfinite was coined by Georg Cantor in 1895. Any finite natural number can be used in at least two ways: as an ordinal and as a cardinal.

Cardinal numbers specify the size of sets (e.g., a bag of five marbles), whereas ordinal numbers specify the order of a member within an ordered set (e.g., "the third man from the left" or "the twenty-seventh day of January"). When extended to transfinite numbers, these two concepts become distinct.

A transfinite cardinal number is used to describe the size of an infinitely large set, while a transfinite ordinal is used to describe the location within an infinitely large set that is ordered. The most notable ordinal and cardinal numbers are, respectively: ω (Omega): the lowest transfinite ordinal number. It is also the order type of the natural numbers under their usual linear ordering. ℵ0 (Aleph-null): the first transfinite cardinal number. It is also the cardinality of the natural numbers. If the axiom of choice holds, the next higher cardinal number is aleph-one, ℵ1. If not, there may be other cardinals which are incomparable with aleph-one and larger than aleph-null.

Either way, there are no cardinals between aleph-null and aleph-one.

The continuum hypothesis is the proposition that there are no intermediate cardinal numbers between ℵ0 and the cardinality of the continuum (the cardinality of the set of real numbers): or equivalently that ℵ1 is the cardinality of the set of real numbers. In Zermelo–Fraenkel set theory, neither the continuum hypothesis nor its negation can be proven. Although transfinite ordinals and cardinals both generalize only the natural numbers, other systems of numbers, including the hyperreal numbers and surreal numbers, provide generalizations of the real numbers.

Actual infinity is to be contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps. As a result, potential infinity is often formalized using the concept of limit. One may write down a symbol, ω, with the verbal description that ‘ω stands for completed (countable) infinity’. This symbol may be added as an ur-element to any set. One may also provide axioms that define addition, multiplication and inequality; specifically, ordinal arithmetic, such that expressions like n<ω can be interpreted as "any natural number is less than completed infinity". Even "common sense" statements such as ω<ω+1 are possible and consistent. The theory is sufficiently well developed, that rather complex algebraic expressions, such as ω^2, ω^ω and even 2^ω can be interpreted as valid algebraic expressions, can be given a verbal description, and can be used in a wide variety of theorems and claims in a consistent and meaningful fashion.

The Absolute Infinite (symbol: Ω) is an extension of the idea of infinity proposed by mathematician Georg Cantor. It can be thought of as a number that is bigger than any conceivable or inconceivable quantity, either finite or transfinite.”

https://en.wikipedia.org/wiki/Infinity https://en.wikipedia.org/wiki/Calculus https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and_the_tortoise https://en.wikipedia.org/wiki/ De_analysi_per_aequationes_numero_terminorum_infinitas https://en.wikipedia.org/wiki/Line_at_infinity https://en.wikipedia.org/wiki/Plane_at_infinity https://en.wikipedia.org/wiki/Point_at_infinity https://en.wikipedia.org/wiki/Beth_number https://en.wikipedia.org/wiki/Transfinite_number https://en.wikipedia.org/wiki/Actual_infinity https://en.wikipedia.org/wiki/Absolute_Infinite

Notes compiled from wikipedia By: Jonathan Barlow Gee

The Lemniscate

“In algebraic geometry, a lemniscate is any of several figure-eight or ∞-shaped curves.

The word comes from the Latin "lēmniscātus" meaning "decorated with ribbons", from the Greek λημνίσκος meaning "ribbons", or which alternatively may refer to the wool from which the ribbons were made. Curves that have been called a lemniscate include three quartic plane curves: the hippopede or lemniscate of Booth, the lemniscate of Bernoulli, and the lemniscate of Gerono. The study of lemniscates (and in particular the hippopede) dates to ancient Greek mathematics, but the term "lemniscate" for curves of this type comes from the work of Jacob Bernoulli in the late 17th century.”

https://en.wikipedia.org/wiki/Lemniscate

On Trans-finites and Number Sets The maximum ordinal (1-d, radius) “natural number” (N) is ∞ (infinity). The maximum cardinal “natural number” is aleph sub-null (transfinite). “Natural” #s include both all “whole” #s (including zero) AND “counting” #s (excluding zero) because these have a 1:1 ratio to one another. (0, 1, 2, …, ∞ = N sub-zero) is equal to the same cardinal sum as (1, 2, 3, …, ∞ = N sub-one) because zero is the empty set and thus adds no measured value.

The maximum (2-d, diameter and area) “integer” (Z) sum is aleph sub-one because it is both +∞ and -∞ and is therefore a quantitatively different sum of “infinity” than the “natural number” set, which is necessarily smaller by exactly 1/2 of (+∞) + (-∞); thus aleph-null is 1/2 the size of aleph-one. Because positive infinity plus negative infinity cancels out to zero, the entirety of the “integer” set is arithmetically reducible (topologically compactable) to zero. An integer maybe either: even (divisible by 2) OR odd, AND either: prime (factorially indivisible) OR composite (having multiple factors).

The set of all positive integers is +∞ and that of all negative integers is -∞, and these combine to form the set of all integers; likewise, the infinite set of all “even” integers and that of all “odd” integers also sum the same quantity, as do the combinations of the infinite set of all “primes” and that of all “composites.”

The maximum (3-d, volume) “rational number” (Q) sum is aleph sub-N (where N is the largest Natural number, AKA aleph sub-zero; thus another expression for aleph sub-N is aleph sub-aleph sub-zero. “Rational” numbers introduce fractions or decimals, making each integer infinitely divisible (linking one to infinity) and adding a “Y” axis to the “X” axis of all integers, increasing the dimensionality of the number set. If measured by decimals, rational numbers may either “terminate” (come to an end after a certain sequence of digits) OR “repeat” (repeating a finite sequence of digits ad infinitum, cf.

1/3 or 33.3-bar %). Thus the sum of all “terminating” rational numbers plus all “repeating” rational numbers equals the maximum rational number sum. The set of all “rational” numbers therefore has a maximum “cardinal” sum (aleph sub-N), however the set of all “irrational” numbers is uncountable and therefore (presumed) to comprise a larger sum than that of all “rational” numbers. Insofar as “rationals” add “depth” to the “number-line” by infinitely expanding every integer into fractional parts, “irrationals” have a maximum cardinal sum of “aleph sub-N+1,” where, no matter how “infinite” the rational number set maybe, the irrational set is more so by a minimum of one additional sum - in the context of irrationals, this “one sum” difference between “rational” and “irrational” number sum sets maybe reducible to the “absolute value of one” (or the circle with radius one surrounding the origin-point at zero).

The maximum (4-d, time) “real” number (R) is thus a combination of both all “rational” and all “irrational” numbers. Therefore, if the maximum sum of all “rational” numbers is reckoned as “aleph sub-N” and that of all “irrational” numbers as “aleph sub-N plus one,” then the maximum “real” number is the sum of both these, expressed as “aleph sub-omega;” this expression is short-hand notation for “aleph sub-N” plus “aleph subN+1” and is synonymous with the concept of “(2 aleph sub-N)+1,” “Beth sub-1” and / or “c” (the “cardinality of the continuum). A “real” number is one whose division into

fractions can continue on in an infinite (unending) sequence post decimal-place. Because of these reasons, the sum of all “real” numbers is considered “uncountable” as it combines multiple “uncountably infinite” sub-sets. The set of all “real” numbers and that of all “imaginary” numbers maybe 1:1. Thus, the (5-d) set of all “complex” numbers (C) that includes both “real” and “imaginary” numbers can be reckoned roughly as “aleph sub-omega squared” (or as the “cardinality of the continuum squared”).

On Indeterminate Forms Because the other 5 usual “indeterminate forms” are transformable (under certain limiting conditions) to the 2 indeterminate forms 0/0 and ∞/∞, and because ∞/∞ maybe expressed as ∞ • ∞^-1, it maybe possible that the indeterminate form 0/0 can also be expressed as (0 • 0)^-1. Also, assuming “complex infinity” (~∞) maybe allowed as one of the variables along with 0, 1 and ∞, then there are (at least) 13 “indeterminate forms,” including complex “duals” for some of the “simple” indeterminate forms: CENTER I = 0/0 (or: (0 • 0)^-1 ?)

LEFT II = ∞ - ∞ III = ~∞ - ~∞ RIGHT IV = 0 • ∞ V = 0 • ~∞ FRONT VI = ∞/∞ (or: ∞ • ∞^-1) VII = ~∞/~∞ BACK VIII = ~∞/∞ IX = ∞/~∞ DOWN X = ∞^0 XI = 0^0 UP XII = 1^~∞ XIII = 1^∞ Because the 6 simple indeterminate forms are all commutable to zero/zero (under certain limiting conditions) it stands to reason the other 6 indeterminate forms (based on “complex infinity”) should be also. Thus, 0/0 maybe the “GCF & LCD” indeterminate.

Some Brief Conclusions

Consider this “hyper-cross” model’s “core” indeterminate form (0/0) alike the ordinal maximum sum of all “natural” numbers (∞); the “left” pair of indeterminate forms (∞ - ∞ and ~∞ - ~∞) alike the cardinal maximum sum for the “naturals” (aleph sub-zero); the “right” pair (0 • ∞ and 0 • ~∞) like the maximum sum of the “integers” (aleph sub-one); the “front” pair (∞/∞ and ~∞/~∞) like the max sum of the “rationals” (being aleph sub-N); the “back” pair (~∞/∞ and ∞/~∞) like that of the max sum of all “irrationals” (being aleph sub-N plus one); the pair “below” (∞^0 and 0^0) like the “cardinality of the continuum” of all “real” numbers (expressed as aleph sub-omega); and the pair “above” (1^∞ and 1^~∞) maybe alike the max sum of all “complex” numbers (aleph sub-omega^2). The reason for placing the “indeterminate forms” in this order and relating them to the maximum sums of each number-set in this sequence is due to the “order of operations”: the “subtractive” pair precedes the “multiplicative,” which precedes the (2 pair of) “divisions” which precede the (2 pair of) “exponential” functions; where, also, 0 < 1 < ∞.

“On the Nature of Indeterminate Forms”

Notes compiled from wikipedia By: Jonathan Barlow Gee