
A number is a
mathematical object used to
count,
measure, and
label. The original examples are the
natural numbers
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 number using a combination of ten fundamental numeric symbols, called
digits. In addition to their use in counting and measuring, numerals are often used for labels (as with
telephone numbers), for ordering (as with
serial numbers), and for codes (as with
ISBNs). In common usage, a ''numeral'' is not clearly distinguished from the ''number'' that it represents.
In
mathematics, the notion of a number has been extended over the centuries to include
0,
negative numbers,
rational numbers such as
one half ,
real numbers such as the
square root of 2 and
, and
complex numbers which extend the real numbers with a
square root of (and its combinations with real numbers by adding or subtracting its multiples).
Calculations 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.
[Gilsdorf, Thomas E. ''Introduction to Cultural Mathematics: With Case Studies in the Otomies and Incas'', John Wiley & Sons, Feb 24, 2012.][Restivo, S. ''Mathematics in Society and History'', Springer Science & Business Media, Nov 30, 1992.] For example, in Western society, the
number 13 is often regarded as
unlucky, and "
a 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.
[Ore, Oystein. ''Number Theory and Its History'', Courier Dover Publications.] 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 (
sets) are considered important special examples of more general categories such as
rings and
fields, and the application of the term "number" is a matter of convention, without fundamental significance.
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. Roman numerals, a system that used combinations of letters from the Roman alphabet, remained dominant in Europe until the spread of the superior
Hindu–Arabic numeral system around the late 14th century, and the Hindu–Arabic numeral system remains the most common system for representing numbers in the world today.
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. 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 (
c. 3400 BC) and the earliest known base 10 system dates to 3100 BC in
Egypt.
Zero
The first known documented use of
zero dates to AD 628, and appeared in the ''
Brāhmasphuṭasiddhānta'', the main work of the
Indian mathematician Brahmagupta. He treated 0 as a number and discussed operations involving it, including
division. By this time (the 7th century) the concept had clearly reached Cambodia as
Khmer numerals, and documentation shows the idea later spreading to
China and the
Islamic world.

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 or 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
algebraic 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 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, meaning ''nothing'', not as a symbol. When division produced 0 as a remainder, , also meaning ''nothing'', was used. These medieval zeros were used by all future medieval
computists (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.
Negative numbers
The abstract concept of negative numbers was recognized as early as 100–50 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. The first reference in a Western work was in the 3rd century AD in
Greece.
Diophantus referred to the equation equivalent to (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 ''
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 ). 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
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
René Descartes did with negative solutions in a
Cartesian coordinate system.
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 used sexagesimal (base 60) fractions with great frequency.
Irrational numbers
The earliest known use of irrational numbers was in the
Indian Sulba Sutras composed between 800 and 500 BC. The first existence proofs of irrational numbers is usually attributed to
Pythagoras, more specifically to the
Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the
square root of 2. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. However, Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but he could not accept irrational numbers, and so, allegedly and frequently reported, he sentenced Hippasus to death by drowning, to impede spreading of this disconcerting news.
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,
Georg Cantor, and
Richard Dedekind 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
cut (Schnitt) in the system of
real numbers, separating all
rational numbers into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass,
Kronecker, and Méray.
The search for roots of
quintic and higher degree equations was an important development, the
Abel–Ruffini theorem (
Ruffini 1799,
Abel 1824) showed that they could not be solved by
radicals (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.
Continued fractions, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of
Euler, 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). Ramus first connected the subject with
determinants, resulting, with the subsequent contributions of Heine,
Möbius, and Günther, in the theory of .
Transcendental numbers and reals
The existence of
transcendental numbers was first established by
Liouville (1844, 1851).
Hermite proved in 1873 that ''e'' is transcendental and
Lindemann proved in 1882 that π 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
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. The symbol
is often used to represent an infinite quantity.
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
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
The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor
Heron of Alexandria in the , 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
Niccolò 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
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
:
seemed capriciously inconsistent with the algebraic identity
:
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
:
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
to guard against this mistake.
The 18th century saw the work of
Abraham de Moivre and
Leonhard Euler.
De Moivre's formula (1730) states:
:
while
Euler's formula of
complex analysis (1748) gave us:
:
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 , where ''a'' and ''b'' are integral, or rational (and ''i'' is one of the two roots of ). His student,
Gotthold Eisenstein, studied the type , where ''ω'' is a complex root of Other such classes (called
cyclotomic fields) 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
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
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 Vallée-Poussin in 1896. Goldbach and Riemann's conjectures remain unproven and unrefuted.
Main classification
Numbers can be classified into
sets, called number systems, such as the
natural numbers and the
real numbers. 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
:
.
Natural numbers

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. 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
, and sometimes
or
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, 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 −7, and . 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
. Here the letter Z comes . The set of integers forms a
ring with the operations addition and multiplication.
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 represents ''m'' parts of a whole divided into ''n'' equal parts. Two different fractions may correspond to the same rational number; for example and are equal, that is:
:
In general,
:
if and only if
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
.
Real numbers
The symbol for the real numbers is R, also written as
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 , 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
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
, the ratio of the
circumference of any circle to its
diameter. When pi is written as
:
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
is irrational. Another well-known number, proven to be an irrational real number, is
:
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 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 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
m. 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 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
.
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 −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
:
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 , 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
.
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 for a suitable integer ''k''. Starting with the first non-negative odd numbers are . Any even number ''m'' has the form where ''k'' is again an
integer. Similarly, the first non-negative even numbers are .
Prime numbers
A prime number, often shortened to just prime, 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.
Goldbach's conjecture is an example of a still unanswered question: "Is every even number the sum of two primes?"
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.
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
μ-recursive functions,
Turing machines or
λ-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
''p''-adic numbers
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.
See also
*
Concrete number
*
List of numbers
*
List of numbers in various languages
*
List of types of numbers
*
*
Complex numbers
*
Numerical cognition
*
Orders of magnitude
*
*
*
*
*
*
Subitizing and counting
Notes
References
*
Tobias Dantzig, ''Number, the language of science; a critical survey written for the cultured non-mathematician'', New York, The Macmillan Company, 1930.
* Erich Friedman,
What's special about this number?'
* Steven Galovich, ''Introduction to Mathematical Structures'', Harcourt Brace Javanovich, 1989, .
*
Paul Halmos, ''Naive Set Theory'', Springer, 1974, .
*
Morris Kline, ''Mathematical Thought from Ancient to Modern Times'', Oxford University Press, 1990.
*
Alfred North Whitehead and
Bertrand Russell, ''
Principia Mathematica'' to *56, Cambridge University Press, 1910.
* Leo Cory, ''A Brief History of Numbers'', Oxford University Press, 2015, .
External links
*
*
BBC Radio 4, In Our Time: Negative Numbers'4000 Years of Numbers' lecture by Robin Wilson, 07/11/07,
Gresham College (available for download as MP3 or MP4, and as a text file).
* ;
On-Line Encyclopedia of Integer Sequences
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