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A number is a mathematical object used to count, measure, and label. The most basic examples are the 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called numerals; for example, "5" is a numeral that represents the number five. 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 Integer using a combination of ten fundamental numeric symbols, called numerical digit. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with ), and for codes (as with ). In common usage, a numeral is not clearly distinguished from the number that it represents.
In mathematics, the notion of number has been extended over the centuries to include zero (0), , such as one half , such as the square root of 2 and , (2025). 9781402002601, Kluwer Academic. ISBN 9781402002601 and which extend the real numbers with a imaginary unit (and its combinations with real numbers by adding or subtracting its multiples). with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, a term which may also refer to number theory, the study of the properties of numbers.
Besides their practical uses, numbers have cultural significance throughout the world. For example, in Western society, the number 13 is often regarded as unlucky, and "One million" may signify "a lot" rather than an exact quantity. Though it is now regarded as pseudoscience, belief in a mystical significance of numbers, known as numerology, permeated ancient and medieval thought. Numerology heavily influenced the development of Greek mathematics, stimulating the investigation of many problems in number theory which are still of interest today.
During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, and may be seen as extending the concept. Among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. In modern mathematics, number systems are considered important special examples of more general algebraic structures such as rings and fields, and the application of the term "number" is a matter of convention, without fundamental significance.Gouvêa, Fernando Q. The Princeton Companion to Mathematics, Chapter II.1, "The Origins of Modern Mathematics", p. 82. Princeton University Press, September 28, 2008. . "Today, it is no longer that easy to decide what counts as a 'number.' The objects from the original sequence of 'integer, rational, real, and complex' are certainly numbers, but so are the p-adics. The quaternions are rarely referred to as 'numbers,' on the other hand, though they can be used to coordinatize certain mathematical notions."
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 earliest unambiguous numbers in the archaeological record are the Mesopotamian base 60 system ( BC); place value emerged in it in the 3rd millennium BCE. The earliest known base 10 system dates to 3100 BC in Egypt.
Brahmagupta's Brāhmasphuṭasiddhānta is the first book that mentions zero as a number, hence Brahmagupta is usually considered the first to formulate the concept of zero. He gave rules of using zero with negative and positive numbers, such as "zero plus a positive number is a positive number, and a negative number plus zero is the negative number". The Brāhmasphuṭasiddhānta is the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number as was done by the Babylonians or as a symbol for a lack of quantity as was done by Ptolemy and the Romans.
The use of 0 as a number should be distinguished from its use as a placeholder numeral in place-value systems. Many ancient texts used 0. Babylonian and Egyptian texts used it. Egyptians used the word nfr to denote zero balance in double entry accounting. 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. In a similar vein, Pāṇini (5th century BC) used the null (zero) operator in the Ashtadhyayi, an early example of an formal grammar for the Sanskrit language (also see Pingala).
There are other uses of zero before Brahmagupta, though the documentation is not as complete as it is in the Brāhmasphuṭasiddhānta.
Records show that the Ancient Greeks seemed unsure about the status of 0 as a number: they asked themselves "How can 'nothing' be something?" leading to interesting philosophical and, by the Medieval period, religious arguments about the nature and existence of 0 and the vacuum. The paradoxes of Zeno of Elea depend in part on the uncertain interpretation of 0. (The ancient Greeks even questioned whether was a number.)
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 but certainly by 40 BC, which became an integral part of Maya numerals and the Maya calendar. Maya arithmetic used base 4 and base 5 written as base 20. George I. Sánchez in 1961 reported a base 4, base 5 "finger" abacus.
By 130 AD, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for 0 (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals. Because it was used alone, not as just a placeholder, this Hellenistic zero was the first documented use of a true zero in the Old World. In later Byzantine Empire manuscripts of his Syntaxis Mathematica ( Almagest), the Hellenistic zero had morphed into the Greek letter Omicron (otherwise meaning 70).
Another true zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus), but as a word, nulla meaning nothing, not as a symbol. When division produced 0 as a remainder, nihil, also meaning nothing, was used. These medieval zeros were used by all future medieval computus (calculators of Easter). An isolated use of their initial, N, was used in a table of Roman numerals by Bede or a colleague about 725, a true zero symbol.
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 Brāhmasphuṭasiddhānta in 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 Flos). René Descartes called them false roots as they cropped up in algebraic polynomials yet he found a way to swap true roots and false roots as well. 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. The first use of negative numbers in a European work was by Nicolas Chuquet during the 15th century. He used them as , 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.
The concept of 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 used sexagesimal (base 60) fractions with great frequency.
The 16th century brought final European acceptance of negative integral and fractional numbers. By the 17th century, mathematicians generally used decimal fractions with modern notation. It was not, however, until the 19th century that mathematicians separated irrationals into algebraic and transcendental parts, and once more undertook the scientific study of irrationals. It had remained almost dormant since Euclid. In 1872, the publication of the theories of Karl Weierstrass (by his pupil E. Kossak), Eduard Heine,Eduard Heine, , Crelle's Journal für die reine und angewandte Mathematik, No. 74 (1872): 172–188. Georg Cantor,Georg Cantor, , Mathematische Annalen, 21, 4 (1883‑12): 545–591. and Richard DedekindRichard Dedekind, Stetigkeit & irrationale Zahlen (Braunschweig: Friedrich Vieweg & Sohn, 1872). Subsequently published in: ———, Gesammelte mathematische Werke, ed. Robert Fricke, Emmy Noether & Öystein Ore (Braunschweig: Friedrich Vieweg & Sohn, 1932), vol. 3, pp. 315–334. was brought about. In 1869, Charles Méray had taken the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method was completely set forth by Salvatore Pincherle (1880), and Dedekind's has received additional prominence through the author's later work (1888) and endorsement by Paul Tannery (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a Dedekind cut in the system of , separating all into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, Kronecker,L. Kronecker, , Crelle's Journal für die reine und angewandte Mathematik, No. 101 (1887): 337–355. and Méray.
The search for roots of Quintic equation and higher degree equations was an important development, the Abel–Ruffini theorem (Ruffini 1799, Abel 1824) showed that they could not be solved by nth root (formulas involving only arithmetical operations and roots). Hence it was necessary to consider the wider set of algebraic numbers (all solutions to polynomial equations). Galois (1832) linked polynomial equations to group theory giving rise to the field of Galois theory.
Simple continued fractions, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler,Leonhard Euler, "Conjectura circa naturam aeris, pro explicandis phaenomenis in atmosphaera observatis", Acta Academiae Scientiarum Imperialis Petropolitanae, 1779, 1 (1779): 162–187. and at the opening of the 19th century were brought into prominence through the writings of Joseph Louis Lagrange. Other noteworthy contributions have been made by Druckenmüller (1837), Kunze (1857), Lemke (1870), and Günther (1872). RamusRamus, "Determinanternes Anvendelse til at bes temme Loven for de convergerende Bröker", in: Det Kongelige Danske Videnskabernes Selskabs naturvidenskabelige og mathematiske Afhandlinger (Kjoebenhavn: 1855), p. 106. first connected the subject with , resulting, with the subsequent contributions of Heine,Eduard Heine, , Crelle's Journal für die reine und angewandte Mathematik, No. 56 (Jan. 1859): 87–99 at 97. Möbius, and Günther,Siegmund Günther, Darstellung der Näherungswerthe von Kettenbrüchen in independenter Form (Erlangen: Eduard Besold, 1873); ———, "Kettenbruchdeterminanten", in: Lehrbuch der Determinanten-Theorie: Für Studirende (Erlangen: Eduard Besold, 1875), c. 6, pp. 156–186. in the theory of Kettenbruchdeterminanten.
Aristotle defined the traditional Western notion of mathematical infinity. He distinguished between actual infinity and potential infinity—the general consensus being that only the latter had true value. Galileo Galilei's Two New Sciences discussed the idea of bijection 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 infinity and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of infinitesimal calculus by Isaac 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.
This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. When René 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
The 18th century saw the work of Abraham de Moivre and Leonhard Euler. De Moivre's formula (1730) states:
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 John Wallis's De algebra tractatus.
In the same year, 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. Gauss studied complex numbers of the form , where a and b are integers (now called ) or rational numbers. His student, Gotthold Eisenstein, studied the type , where ω is a complex root of (now called Eisenstein integers). Other such classes (called ) of complex numbers derive from the roots of unity for higher values of k. This generalization is largely due to Ernst Kummer, who also invented , 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.
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 Vallée-Poussin in 1896. Goldbach and Riemann's conjectures remain unproven and unrefuted.
| + Main number systems !Symbol !Name !Examples/Explanation |
Each of these number systems is a subset of the next one. So, for example, a rational number is also a real number, and every real number is also a complex number. This can be expressed symbolically as
A more complete list of number sets appears in the following diagram.
In the base 10 numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten numerical digit: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The Radix 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, 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.
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.
In general,
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 −7 can be written . The symbol for the rational numbers is Q (for quotient), also written Blackboard bold
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 , 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 can be written as 0.333..., with an ellipsis to indicate that the pattern continues. Forever repeating 3s are also written as 0..
It turns out that these repeating decimals (including the Trailing zero) 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 , the ratio of the circumference of any circle to its diameter. When pi is written as
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 rounding or truncation 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 and strictly less than (rounding to 3 decimals), or of any real number greater or equal to and strictly less than (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 Metre. If the sides of a rectangle are measured as 1.23 m and 4.56 m, then multiplication gives an area for the rectangle between and . 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 9s, 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 0s can be rewritten by dropping the 0s to the right of the rightmost nonzero digit, and a decimal numeral with an unlimited number of 9s can be rewritten by increasing by one the rightmost digit less than 9, and changing all the 9s to the right of that digit to 0s. Finally, an unlimited sequence of 0s to the right of a 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 9s, 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 .
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 space, but unlike the real numbers, it is not total order. 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 ordered field.
One answered question, as to whether every integer greater than one is a product of primes in only one way, except for a rearrangement of the primes, was confirmed; this proven claim is called the fundamental theorem of arithmetic. A proof appears in Euclid's Elements.
The periods can be extended by permitting the integrand to be the product of an algebraic function and the exponential of an algebraic function. This gives another countable ring: the exponential periods. The number e as well as Euler's constant are exponential periods.
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.
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.
where the coefficients , , , are real numbers, and , are 3 different imaginary units.
Each hypercomplex number system is a subset of the next hypercomplex number system of double dimensions obtained via the Cayley–Dickson construction. For example, the 4-dimensional quaternions are a subset of the 8-dimensional quaternions , which are in turn a subset of the 16-dimensional sedenions , in turn a subset of the 32-dimensional , and ad infinitum with dimensions, with n being any non-negative integer. Including the complex and real numbers and their subsets, this can be expressed symbolically as:
Alternatively, starting from the real numbers , which have zero complex units, this can be expressed as
Superreal number and extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form fields.
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