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History

Numerals

Numbers should be distinguished from numerals, the symbols used to represent numbers. The Egyptians invented the first ciphered numeral system, and the Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets.JOURNAL, Chrisomalis, Stephen, 2003-09-01, The Egyptian origin of the Greek alphabetic numerals,weblink Antiquity, 77, 297, 485â€“96, 10.1017/S0003598X00092541, 0003-598X, Roman numerals, a system that used combinations of letters from the Roman alphabet, remained dominant in Europe until the spread of the superior Hinduâ€“Arabic numeral system around the late 14th century, and the Hinduâ€“Arabic numeral system remains the most common system for representing numbers in the world today.BOOK,weblink The Earth and Its Peoples: A Global History, Volume 1, Crossley, Pamela, Headrick,, Daniel, Hirsch, Steven, Johnson, Lyman, Cengage Learning, 2010, 1-4390-8474-2, 192, Indian mathematicians invented the concept of zero and developed the "Arabic" numerals and system of place-value notation used in most parts of the world today, Richard, Bulliet, {{better source|date=January 2017}} The key to the effectiveness of the system was the symbol for zero, which was developed by ancient Indian mathematicians around 500 AD.

First use of numbers

Bones and other artifacts have been discovered with marks cut into them that many believe are tally marks.Marshak, A., The Roots of Civilisation; Cognitive Beginnings of Manâ€™s First Art, Symbol and Notation, (Weidenfeld & Nicolson, London: 1972), 81ff. These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals.A tallying system has no concept of place value (as in modern decimal notation), which limits its representation of large numbers. Nonetheless tallying systems are considered the first kind of abstract numeral system.The first known system with place value was the Mesopotamian base 60 system (ca. 3400 BC) and the earliest known base 10 system dates to 3100 BC in Egypt.WEB,weblink Egyptian Mathematical Papyri â€“ Mathematicians of the African Diaspora, Math.buffalo.edu, 2012-01-30,

Negative numbers {{anchor|History of negative numbers}}

{{further|History of negative numbers}}The abstract concept of negative numbers was recognized as early as 100â€“50 BC in China. The Nine Chapters on the Mathematical Art contains methods for finding the areas of figures; red rods were used to denote positive coefficients, black for negative.BOOK, Staszkow, Ronald, Robert Bradshaw, The Mathematical Palette (3rd ed.), Brooks Cole, 2004, 41, 0-534-40365-4, The first reference in a Western work was in the 3rd century AD in Greece. Diophantus referred to the equation equivalent to {{nowrap|4x + 20 {{=}} 0}} (the solution is negative) in Arithmetica, saying that the equation gave an absurd result.During the 600s, negative numbers were in use in India to represent debts. Diophantus' previous reference was discussed more explicitly by Indian mathematician Brahmagupta, in BrÄhmasphuá¹­asiddhÄnta 628, who used negative numbers to produce the general form quadratic formula that remains in use today. However, in the 12th century in India, Bhaskara gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots."European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debts (chapter 13 of Liber Abaci, 1202) and later as losses (in ). At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit of the corresponding positive number's numeral.BOOK, Smith, David Eugene, David_Eugene_Smith, History of Modern Mathematics, Dover Publications, 1958, 259, 0-486-20429-4, The first use of negative numbers in a European work was by Nicolas Chuquet during the 15th century. He used them as exponents, but referred to them as "absurd numbers".As recently as the 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless, just as RenÃ© Descartes did with negative solutions in a Cartesian coordinate system.

Rational numbers {{anchor|History of rational numbers}}

It is likely that the concept of fractional numbers dates to prehistoric times. The Ancient Egyptians used their Egyptian fraction notation for rational numbers in mathematical texts such as the Rhind Mathematical Papyrus and the Kahun Papyrus. Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of number theory. The best known of these is Euclid's Elements, dating to roughly 300  BC. Of the Indian texts, the most relevant is the Sthananga Sutra, which also covers number theory as part of a general study of mathematics.The concept of decimal fractions is closely linked with decimal place-value notation; the two seem to have developed in tandem. For example, it is common for the Jain math sutra to include calculations of decimal-fraction approximations to pi or the square root of 2. Similarly, Babylonian math texts had always used sexagesimal (base 60) fractions with great frequency.

Transcendental numbers and reals {{anchor|History of transcendental numbers and reals}}

{{further|History of Ï€}}The existence of transcendental numbersWEB, Bogomolny, A., Cut-the-Knot, What's a number?, Interactive Mathematics Miscellany and Puzzles,weblink 11 July 2010, was first established by Liouville (1844, 1851). Hermite proved in 1873 that e is transcendental and Lindemann proved in 1882 that Ï€ is transcendental. Finally, Cantor showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite, so there is an uncountably infinite number of transcendental numbers.

Infinity and infinitesimals {{anchor|History of infinity and infinitesimals}}

{{further|History of infinity}}The earliest known conception of mathematical infinity appears in the Yajur Veda, an ancient Indian script, which at one point states, "If you remove a part from infinity or add a part to infinity, still what remains is infinity." Infinity was a popular topic of philosophical study among the Jain mathematicians c. 400 BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.Aristotle defined the traditional Western notion of mathematical infinity. He distinguished between actual infinity and potential infinityâ€”the general consensus being that only the latter had true value. Galileo Galilei's Two New Sciences discussed the idea of one-to-one correspondences between infinite sets. But the next major advance in the theory was made by Georg Cantor; in 1895 he published a book about his new set theory, introducing, among other things, transfinite numbers and formulating the continuum hypothesis.In the 1960s, Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of infinitesimal calculus by Newton and Leibniz.A modern geometrical version of infinity is given by projective geometry, which introduces "ideal points at infinity", one for each spatial direction. Each family of parallel lines in a given direction is postulated to converge to the corresponding ideal point. This is closely related to the idea of vanishing points in perspective drawing.

Complex numbers {{anchor|History of complex numbers}}

{{further|History of complex numbers}}The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor Heron of Alexandria in the {{nowrap|1st century AD}}, when he considered the volume of an impossible frustum of a pyramid. They became more prominent when in the 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians such as NiccolÃ² Fontana Tartaglia and Gerolamo Cardano. It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers.This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. When RenÃ© Descartes coined the term "imaginary" for these quantities in 1637, he intended it as derogatory. (See imaginary number for a discussion of the "reality" of complex numbers.) A further source of confusion was that the equation
left ( sqrt{-1}right )^2 =sqrt{-1}sqrt{-1}=-1
seemed capriciously inconsistent with the algebraic identity
sqrt{a}sqrt{b}=sqrt{ab},
which is valid for positive real numbers a and b, and was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity, and the related identity
frac{1}{sqrt{a}}=sqrt{frac{1}{a}}
in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led him to the convention of using the special symbol i in place of sqrt{-1} to guard against this mistake.The 18th century saw the work of Abraham de Moivre and Leonhard Euler. De Moivre's formula (1730) states:
(cos theta + isin theta)^{n} = cos n theta + isin n theta
while Euler's formula of complex analysis (1748) gave us:
cos theta + isin theta = e ^{itheta }.
The existence of complex numbers was not completely accepted until Caspar Wessel described the geometrical interpretation in 1799. Carl Friedrich Gauss rediscovered and popularized it several years later, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De Algebra tractatus.Also in 1799, Gauss provided the first generally accepted proof of the fundamental theorem of algebra, showing that every polynomial over the complex numbers has a full set of solutions in that realm. The general acceptance of the theory of complex numbers is due to the labors of Augustin Louis Cauchy and Niels Henrik Abel, and especially the latter, who was the first to boldly use complex numbers with a success that is well known.Gauss studied complex numbers of the form {{nowrap|a + bi}}, where a and b are integral, or rational (and i is one of the two roots of {{nowrap|x2 + 1 {{=}} 0}}). His student, Gotthold Eisenstein, studied the type {{nowrap|a + bÏ‰}}, where Ï‰ is a complex root of {{nowrap|x3 âˆ’ 1 {{=}} 0.}} Other such classes (called cyclotomic fields) of complex numbers derive from the roots of unity {{nowrap|xk âˆ’ 1 {{=}} 0}} for higher values of k. This generalization is largely due to Ernst Kummer, who also invented ideal numbers, which were expressed as geometrical entities by Felix Klein in 1893.In 1850 Victor Alexandre Puiseux took the key step of distinguishing between poles and branch points, and introduced the concept of essential singular points. This eventually led to the concept of the extended complex plane.

Prime numbers {{anchor|History of prime numbers}}

Prime numbers have been studied throughout recorded history. Euclid devoted one book of the Elements to the theory of primes; in it he proved the infinitude of the primes and the fundamental theorem of arithmetic, and presented the Euclidean algorithm for finding the greatest common divisor of two numbers.In 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to the Renaissance and later eras.In 1796, Adrien-Marie Legendre conjectured the prime number theorem, describing the asymptotic distribution of primes. Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges, and the Goldbach conjecture, which claims that any sufficiently large even number is the sum of two primes. Yet another conjecture related to the distribution of prime numbers is the Riemann hypothesis, formulated by Bernhard Riemann in 1859. The prime number theorem was finally proved by Jacques Hadamard and Charles de la VallÃ©e-Poussin in 1896. Goldbach and Riemann's conjectures remain unproven and unrefuted.

{{anchor|Classification|Classification of numbers}}Main classification

{{Redirect|Number system|systems for expressing numbers|Numeral system}}{{See also|List of types of numbers}}Numbers can be classified into sets, called number systems, such as the natural numbers and the real numbers."Eine Menge, ist die Zusammenfassung bestimmter, wohlunterschiedener Objekte unserer Anschauung oder unseres Denkens â€“ welche Elemente der Menge genannt werden â€“ zu einem Ganzen."  The major categories of numbers are as follows:{|class="wikitable" style="text-align: center; width: 400px; height: 200px;"|+ Main number systems!mathbb{N}!Natural| 0, 1, 2, 3, 4, 5, ... or 1, 2, 3, 4, 5, ...mathbb{N}_0 or mathbb{N}_1 are sometimes used.
!mathbb{Z}!Integer|..., âˆ’5, âˆ’4, âˆ’3, âˆ’2, âˆ’1, 0, 1, 2, 3, 4, 5, ...
!mathbb{Q}!Rational
a|b}} where a and b are integers and b is not 0
!mathbb{R}!Real|The limit of a convergent sequence of rational numbers
!mathbb{C}!Complex|a + bi where a and b are real numbers and i is a formal square root of âˆ’1
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.{{Citation needed|date=June 2017}} The resulting hierarchy allows, for example, to talk, formally correctly, about real numbers that are rational numbers, and is expressed symbolically by writing
mathbb{N} subset mathbb{Z} subset mathbb{Q} subset mathbb{R} subset mathbb{C}.

Natural numbers

(File:Nat num.svg|thumb|The natural numbers, starting with 1)The most familiar numbers are the natural numbers (sometimes called whole numbers or counting numbers): 1, 2, 3, and so on. Traditionally, the sequence of natural numbers started with 1 (0 was not even considered a number for the Ancient Greeks.) However, in the 19th century, set theorists and other mathematicians started including 0 (cardinality of the empty set, i.e. 0 elements, where 0 is thus the smallest cardinal number) in the set of natural numbers.{{MathWorld|title=Natural Number|id=NaturalNumber}}{{Citation |url=http://www.merriam-webster.com/dictionary/natural%20number |title=natural number |work=Merriam-Webster.com |publisher=Merriam-Webster |accessdate=4 October 2014}}
Today, different mathematicians use the term to describe both sets, including 0 or not. The mathematical symbol for the set of all natural numbers is N, also written mathbb{N}, and sometimes mathbb{N}_0 or mathbb{N}_1 when it is necessary to indicate whether the set should start with 0 or 1, respectively.
In the base 10 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. The radix or base is the number of unique numerical digits, including zero, that a numeral system uses to represent numbers (for the decimal system, the radix is 10). In this base 10 system, the rightmost digit of a natural number has a place value of 1, and every other digit has a place value ten times that of the place value of the digit to its right.In set theory, which is capable of acting as an axiomatic foundation for modern mathematics,BOOK, Suppes, Patrick, Patrick_Suppes, Axiomatic Set Theory, Courier Dover Publications, 1972, 1, 0-486-61630-4,weblink natural numbers can be represented by classes of equivalent sets. For instance, the number 3 can be represented as the class of all sets that have exactly three elements. Alternatively, in Peano Arithmetic, the number 3 is represented as sss0, where s is the "successor" function (i.e., 3 is the third successor of 0). Many different representations are possible; all that is needed to formally represent 3 is to inscribe a certain symbol or pattern of symbols three times.

Integers

The negative of a positive integer is defined as a number that produces 0 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 written âˆ’7, and {{nowrap|7 + (âˆ’7) {{=}} 0}}. When the set of negative numbers is combined with the set of natural numbers (including 0), the result is defined as the set of integers, Z also written mathbb{Z}. Here the letter Z comes {{ety|de|Zahl|number}}. The set of integers forms a ring with the operations addition and multiplication.{{Mathworld|Integer|Integer}}The natural numbers form a subset of the integers. As there is no common standard for the inclusion or not of zero in the natural numbers, the natural numbers without zero are commonly referred to as positive integers, and the natural numbers with zero are referred to as non-negative integers.

Rational numbers

A rational number is a number that can be expressed as a fraction with an integer numerator and a positive integer denominator. Negative denominators are allowed, but are commonly avoided, as every rational number is equal to a fraction with positive denominator. Fractions are written as two integers, the numerator and the denominator, with a dividing bar between them. The fraction {{sfrac|m|n}} represents m parts of a whole divided into n equal parts. Two different fractions may correspond to the same rational number; for example {{sfrac|1|2}} and {{sfrac|2|4}} are equal, that is:
{1 over 2} = {2 over 4}.
In general,
{a over b} = {c over d} if and only if { a times d} = {c times b}.
If the absolute value of m is greater than n (supposed to be positive), then the absolute value of the fraction is greater than 1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or 0. The set of all rational numbers includes the integers since every integer can be written as a fraction with denominator 1. For example âˆ’7 can be written {{sfrac|âˆ’7|1}}. The symbol for the rational numbers is Q (for quotient), also written mathbb{Q}.

Real numbers

The symbol for the real numbers is R, also written as mathbb{R}. They include all the measuring numbers. Every real number corresponds to a point on the number line. The following paragraph will focus primarily on positive real numbers. The treatment of negative real numbers is according to the general rules of arithmetic and their denotation is simply prefixing the corresponding positive numeral by a minus sign, e.g. -123.456.Most real numbers can only be approximated by decimal numerals, in which a decimal point is placed to the right of the digit with place value 1. Each digit to the right of the decimal point has a place value one-tenth of the place value of the digit to its left. For example, 123.456 represents {{sfrac|123456|1000}}, or, in words, one hundred, two tens, three ones, four tenths, five hundredths, and six thousandths. A real number can be expressed by a finite number of decimal digits only if it is rational and its fractional part has a denominator whose prime factors are 2 or 5 or both, because these are the prime factors of 10, the base of the decimal system. Thus, for example, one half is 0.5, one fifth is 0.2, one-tenth is 0.1, and one fiftieth is 0.02. Representing other real numbers as decimals would require an infinite sequence of digits to the right of the decimal point. If this infinite sequence of digits follows a pattern, it can be written with an ellipsis or another notation that indicates the repeating pattern. Such a decimal is called a repeating decimal. Thus {{sfrac|3}} can be written as 0.333..., with an ellipsis to indicate that the pattern continues. Forever repeating 3s are also written as 0.{{overline|3}}.It turns out that these repeating decimals (including the repetition of zeroes) denote exactly the rational numbers, i.e., all rational numbers are also real numbers, but it is not the case that every real number is rational. A real number that is not rational is called irrational. A famous irrational real number is the number {{pi}}, the ratio of the circumference of any circle to its diameter. When pi is written as
pi = 3.14159265358979dots,
as it sometimes is, the ellipsis does not mean that the decimals repeat (they do not), but rather that there is no end to them. It has been proved that {{pi}} is irrational. Another well-known number, proven to be an irrational real number, is
sqrt{2} = 1.41421356237dots,
the square root of 2, that is, the unique positive real number whose square is 2. Both these numbers have been approximated (by computer) to trillions {{nowrap|( 1 trillion {{=}} 1012 {{=}} 1,000,000,000,000 )}} of digits.Not only these prominent examples but almost all real numbers are irrational and therefore have no repeating patterns and hence no corresponding decimal numeral. They can only be approximated by decimal numerals, denoting rounded or truncated real numbers. Any rounded or truncated number is necessarily a rational number, of which there are only countably many. All measurements are, by their nature, approximations, and always have a margin of error. Thus 123.456 is considered an approximation of any real number greater or equal to {{sfrac|1234555|10000}} and strictly less than {{sfrac|1234565|10000}} (rounding to 3 decimals), or of any real number greater or equal to {{sfrac|123456|1000}} and strictly less than {{sfrac|123457|1000}} (truncation after the 3. decimal). Digits that suggest a greater accuracy than the measurement itself does, should be removed. The remaining digits are then called significant digits. For example, measurements with a ruler can seldom be made without a margin of error of at least 0.001  meters. If the sides of a rectangle are measured as 1.23 meters and 4.56 meters, then multiplication gives an area for the rectangle between {{nowrap|5.614591 square meters}} and {{nowrap|5.603011 square meters}}. Since not even the second digit after the decimal place is preserved, the following digits are not significant. Therefore, the result is usually rounded to 5.61.Just as the same fraction can be written in more than one way, the same real number may have more than one decimal representation. For example, 0.999..., 1.0, 1.00, 1.000, ..., all represent the natural number 1. A given real number has only the following decimal representations: an approximation to some finite number of decimal places, an approximation in which a pattern is established that continues for an unlimited number of decimal places or an exact value with only finitely many decimal places. In this last case, the last non-zero digit may be replaced by the digit one smaller followed by an unlimited number of 9's, or the last non-zero digit may be followed by an unlimited number of zeros. Thus the exact real number 3.74 can also be written 3.7399999999... and 3.74000000000.... Similarly, a decimal numeral with an unlimited number of 0's can be rewritten by dropping the 0's to the right of the decimal place, and a decimal numeral with an unlimited number of 9's can be rewritten by increasing the rightmost -9 digit by one, changing all the 9's to the right of that digit to 0's. Finally, an unlimited sequence of 0's to the right of the decimal place can be dropped. For example, 6.849999999999... = 6.85 and 6.850000000000... = 6.85. Finally, if all of the digits in a numeral are 0, the number is 0, and if all of the digits in a numeral are an unending string of 9's, you can drop the nines to the right of the decimal place, and add one to the string of 9s to the left of the decimal place. For example, 99.999... = 100.The real numbers also have an important but highly technical property called the least upper bound property.It can be shown that any ordered field, which is also complete, is isomorphic to the real numbers. The real numbers are not, however, an algebraically closed field, because they do not include a solution (often called a square root of minus one) to the algebraic equation x^2+1=0.

Complex numbers

Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers. This set of numbers arose historically from trying to find closed formulas for the roots of cubic and quadratic polynomials. This led to expressions involving the square roots of negative numbers, and eventually to the definition of a new number: a square root of âˆ’1, denoted by i, a symbol assigned by Leonhard Euler, and called the imaginary unit. The complex numbers consist of all numbers of the form
,a + b i
where a and b are real numbers. Because of this, complex numbers correspond to points on the complex plane, a vector space of two real dimensions. In the expression {{nowrap|a + bi}}, the real number a is called the real part and b is called the imaginary part. If the real part of a complex number is 0, then the number is called an imaginary number or is referred to as purely imaginary; if the imaginary part is 0, then the number is a real number. Thus the real numbers are a subset of the complex numbers. If the real and imaginary parts of a complex number are both integers, then the number is called a Gaussian integer. The symbol for the complex numbers is C or mathbb{C}.The fundamental theorem of algebra asserts that the complex numbers form an algebraically closed field, meaning that every polynomial with complex coefficients has a root in the complex numbers. Like the reals, the complex numbers form a field, which is complete, but unlike the real numbers, it is not ordered. That is, there is no consistent meaning assignable to saying that I is greater than 1, nor is there any meaning in saying that I is less than 1. In technical terms, the complex numbers lack a total order that is compatible with field operations.

Subclasses of the integers

Even and odd numbers

An even number is an integer that is "evenly divisible" by two, that is divisible by two without remainder; an odd number is an integer that is not even. (The old-fashioned term "evenly divisible" is now almost always shortened to "divisible".) Any odd number n may be constructed by the formula {{nowrap|n {{=}} 2k + 1,}} for a suitable integer k. Starting with {{nowrap|k {{=}} 0,}} the first non-negative odd numbers are {1, 3, 5, 7, ...}. Any even number m has the form {{nowrap|m {{=}} 2k}} where k is again an integer. Similarly, the first non-negative even numbers are {0, 2, 4, 6, ...}.

Prime numbers

A prime number is an integer greater than 1 that is not the product of two smaller positive integers. The first few prime numbers are 2, 3, 5, 7, and 11. There is no such simple formula as for odd and even numbers to generate the prime numbers. The primes have been widely studied for more than 2000 years and have led to many questions, only some of which have been answered. The study of these questions belongs to number theory. An example of a still unanswered question is, whether every even number is the sum of two primes. This is called Goldbach's conjecture.The question, whether every integer greater than one is a product of primes in only one way, except for a rearrangement of the primes, has been answered to the positive: this proven claim is called fundamental theorem of arithmetic. A proof appears in Euclid's Elements.

Other classes of integers

Many subsets of the natural numbers have been the subject of specific studies and have been named, often after the first mathematician that has studied them. Example of such sets of integers are Fibonacci numbers and perfect numbers. For more examples, see Integer sequence.

Subclasses of the complex numbers

Algebraic, irrational and transcendental numbers

Algebraic numbers are those that are a solution to a polynomial equation with integer coefficients. Real numbers that are not rational numbers are called irrational numbers. Complex numbers which are not algebraic are called transcendental numbers. The algebraic numbers that are solutions of a monic polynomial equation with integer coefficients are called algebraic integers.

Constructible numbers

Motivated by the classical problems of constructions with straightedge and compass, the constructible numbers are those complex numbers whose real and imaginary parts can be constructed using straightedge and compass, starting from a given segment of unit length, in a finite number of steps.

Computable numbers

A computable number, also known as recursive number, is a real number such that there exists an algorithm which, given a positive number n as input, produces the first n digits of the computable number's decimal representation. Equivalent definitions can be given using Î¼-recursive functions, Turing machines or Î»-calculus. The computable numbers are stable for all usual arithmetic operations, including the computation of the roots of a polynomial, and thus form a real closed field that contains the real algebraic numbers.The computable numbers may be viewed as the real numbers that may be exactly represented in a computer: a computable number is exactly represented by its first digits and a program for computing further digits. However, the computable numbers are rarely used in practice. One reason is that there is no algorithm for testing the equality of two computable numbers. More precisely, there cannot exist any algorithm which takes any computable number as an input, and decides in every case if this number is equal to zero or not.The set of computable numbers has the same cardinality as the natural numbers. Therefore, almost all real numbers are non-computable. However, it is very difficult to produce explicitly a real number that is not computable.

Extensions of the concept

The p-adic numbers may have infinitely long expansions to the left of the decimal point, in the same way that real numbers may have infinitely long expansions to the right. The number system that results depends on what base is used for the digits: any base is possible, but a prime number base provides the best mathematical properties. The set of the p-adic numbers contains the rational numbers, but is not contained in the complex numbers.The elements of an algebraic function field over a finite field and algebraic numbers have many similar properties (see Function field analogy). Therefore, they are often regarded as numbers by number theorists. The p-adic numbers play an important role in this analogy.

Hypercomplex numbers

Some number systems that are not included in the complex numbers may be constructed from the real numbers in a way that generalize the construction of the complex numbers. They are sometimes called hypercomplex numbers. They include the quaternions H, introduced by Sir William Rowan Hamilton, in which multiplication is not commutative, the octonions, in which multiplication is not associative in addition to not being commutative, and the sedenions, in which multiplication is not alternative, neither associative nor commutative.

Transfinite numbers

For dealing with infinite sets, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former gives the ordering of the set, while the latter gives its size. For finite sets, both ordinal and cardinal numbers are identified with the natural numbers. In the infinite case, many ordinal numbers correspond to the same cardinal number.

Nonstandard numbers

Hyperreal numbers are used in non-standard analysis. The hyperreals, or nonstandard reals (usually denoted as *R), denote an ordered field that is a proper extension of the ordered field of real numbers R and satisfies the transfer principle. This principle allows true first-order statements about R to be reinterpreted as true first-order statements about *R.Superreal and surreal numbers extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form fields.

{{reflist}}

References

• Tobias Dantzig, Number, the language of science; a critical survey written for the cultured non-mathematician, New York, The Macmillan company, 1930.
• Steven Galovich, Introduction to Mathematical Structures, Harcourt Brace Javanovich, 23 January 1989, {{isbn|0-15-543468-3}}.
• Paul Halmos, Naive Set Theory, Springer, 1974, {{isbn|0-387-90092-6}}.
• Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972.
• Alfred North Whitehead and Bertrand Russell, Principia Mathematica to 56, Cambridge University Press, 1910.
• George I. Sanchez, Arithmetic in Maya, Austin-Texas, 1961

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