# Dictionary Definition

number

### Noun

1 the property possessed by a sum or total or
indefinite quantity of units or individuals; "he had a number of
chores to do"; "the number of parameters is small"; "the figure was
about a thousand" [syn: figure]

2 a concept of quantity derived from zero and
units; "every number has a unique position in the sequence"

3 a short theatrical performance that is part of
a longer program; "he did his act three times every evening"; "she
had a catchy little routine"; "it was one of the best numbers he
ever did" [syn: act,
routine, turn, bit]

4 a numeral or string of numerals that is used
for identification; "she refused to give them her Social Security
number" [syn: identification
number]

5 the number is used in calling a particular
telephone; "he has an unlisted number" [syn: phone
number, telephone
number]

6 a symbol used to represent a number; "he
learned to write the numerals before he went to school" [syn:
numeral]

7 one of a series published periodically; "she
found an old issue of the magazine in her dentist's waitingroom"
[syn: issue]

8 a select company of people; "I hope to become
one of their number before I die"

9 the grammatical category for the forms of nouns
and pronouns and verbs that are used depending on the number of
entities involved (singular or dual or plural); "in English the
subject and the verb must agree in number"

10 an item of merchandise offered for sale; "she
preferred the black nylon number"; "this sweater is an all-wool
number"

11 a clothing measurement; "a number 13
shoe"

### Verb

1 add up in number or quantity; "The bills
amounted to $2,000"; "The bill came to $2,000" [syn: total, add up, come, amount]

2 give numbers to; "You should number the pages
of the thesis"

3 enumerate; "We must number the names of the
great mathematicians" [syn: list]

4 put into a group; "The academy counts several
Nobel Prize winners among its members" [syn: count]

5 determine the number or amount of; "Can you
count the books on your shelf?"; "Count your change" [syn: count, enumerate, numerate]

6 place a limit on the number of [syn: keep
down]

# User Contributed Dictionary

## English

### Etymology 1

#### Noun

- An abstract entity used to describe quantity.
- A symbol for a non-negative integer; a numeral.
- A member of one of several classes: natural
numbers,integers,
rational
numbers, real numbers,
complex
numbers, quaternions.
- The equation e^+1=0 includes the most important numbers: 1, 0, \pi, i, and e.

- (Followed by a numeral; used attributive) Showing the rank of something in a
list or sequence. Abbreviations: No or No., no or no. (in each case,
sometimes written with a superscript "o", like Nº or №).
The symbol "#" is also used in this manner.
- Horse number 5 won the race.

- Quantity.
- Any number of people can be reading from a given repository at a time

- Of a word or phrase, the state of being singular, dual or plural, shown by inflection.
- A performance;
especially, a single song
or song and dance routine within a larger show.
- For his second number, he sang "The Moon Shines Bright".

##### Synonyms

##### Derived terms

- abundant number
- algebraic number
- binary number
- cardinal number
- complex number
- decimal number
- deficient number
- hexadecimal number
- house number
- hyperreal number
- hypercomplex number
- imaginary number
- irrational number
- natural number
- number-cruncher
- number-crunching
- number field
- number line
- number one
- number two
- number theory
- ordinal number
- opposite number
- perfect number
- phone number
- prime number
- rational number
- real number
- round number
- serial number
- surreal number
- telephone number
- transcendental number
- transfinite number
- whole number
- without number

##### Related terms

##### Translations

abstract entity

- Arabic:
- Basque: lehen?
- Belarusian: лік (lik)
- Breton: niver
- Bulgarian: число
- Chinese: 数 (shù), 数目 (shùmù)
- Croatian: broj
- Czech: číslo
- Danish: tal
- Dutch: aantal, hoeveelheid
- Esperanto: numero
- Estonian: arv
- Finnish: luku, numero
- French: nombre, chiffre
- Galician: número
- Georgian: რიცხვი (ritsχvi)
- German: Anzahl, Zahl
- Greek: αριθμός, πλήθος
- Hebrew: מספר
- Hungarian: szám
- Icelandic: tala
- Ido: numbro
- Indonesian: angka, bilangan
- Interlingua: numero
- Irish: uimhir
- Italian: numero
- Japanese: 数 (かず, kazu), 数字 (すうじ, sūji)
- Korean: 수 (su)
- Kurdish:
- Sorani: ژماره

- Latin: numero, numerus
- Lithuanian: skaičiai
- Malay: nombor, angka
- Maltese: numru
- Nepali: (sankhya, anka)
- Norwegian: antall, tall
- Novial: nombre
- Old English: rīm, ġerīm , tæl
- Polish: liczba
- Portuguese: número
- Romanian: număr
- Russian: число
- Serbian: broj , čislo
- Sicilian: nummuru, nùmmuru
- Slovak: číslo
- Slovene: število
- Spanish: número
- Swedish: tal
- Turkish: sayı

numeral

- Croatian: broj
- Finnish: numeraali
- French: chiffre
- Greek: αριθμός, ψηφίο
- Icelandic: tölustafur
- Indonesian: nomor
- Polish: cyfra
- Portuguese: número
- Russian: число, цифра

mathematical number

used to show the rank of something in a list or
sequence

quantity

grammar: state of being singular, dual or plural

- Croatian: broj
- French: nombre
- Greek: αριθμός
- Hungarian: szám
- Icelandic: tala
- Latin: numerus
- Polish: liczba
- Portuguese: número
- Russian: число

#### Verb

##### Translations

label with numbers; assign numbers to

- Breton: niverenniñ
- Dutch: nummeren
- Finnish: numeroida
- French: numéroter
- German: nummerieren
- Greek: αριθμώ
- Hungarian: megszámoz
- Icelandic: númera, tölusetja
- Interlingua: numerar
- Italian: numerare
- Latin: numerare
- Portuguese: numerar
- Romanian: număra
- Russian: нумеровать (numerovát’)
- Spanish: numerar
- Swedish: numrera

to total; to amount to

- German: zählen
- Hungarian: kitesz egy mennyiséget, valamennyit számlál
- Russian: насчитывать (nasčítyvat'), исчисляться (isčislját’s’a)

#### See also

### Etymology 2

From numb + -er.#### Adjective

number- comparative of numb

# Extensive Definition

A number is an abstract
object, tokens
of which are symbols used
in counting and
measuring. A symbol
which represents a number is called a numeral, but
in common usage the word number is used for both the abstract
object and the symbol. In addition to their use in counting and
measuring, numerals are often used for labels (telephone
numbers), for ordering (serial
numbers), and for codes (ISBNs). In mathematics, the definition
of number has been extended over the years to include such numbers
as zero, negative
numbers, rational
numbers, irrational
numbers, and complex
numbers. As a result, there is no one encompassing definition
of number and the concept of number is open for further
development.

Certain procedures which input one or more
numbers and output a number are called numerical operations.
Unary
operations input a single number and output a single number.
For example, the successor operation adds one to an integer: the
successor of 4 is 5. More common are binary
operations which input two numbers and output a single number.
Examples of binary operations include addition, subtraction, multiplication, division,
and exponentiation. The study
of numerical operations is called arithmetic.

The branch of mathematics that studies
structures of number systems such as groups,
rings and
fields
is called abstract
algebra.

## Types of numbers

Numbers can be classified into sets, called number systems. (For different methods of expressing numbers with symbols, such as the Roman numerals, see numeral systems.)### Natural numbers

The most familiar numbers are the natural numbers or counting numbers: one, two, three, ... . Some people also include zero in the natural numbers; however, others do not.In the base ten number
system, in almost universal use today for arithmetic operations,
the symbols for natural numbers are written using ten digits:
0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In this base ten system, the
rightmost digit of a natural number has a place value of one, and
every other digit has a place value ten times that of the place
value of the digit to its right. The symbol for the set of all
natural numbers is N, also written \mathbb.

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. Many different representations are possible;
all that is needed to formally represent 3 is to inscribe a certain
symbol or pattern of symbols 3 times.

### Integers

Negative numbers are numbers that are less than zero. They are the opposite of positive numbers. For example, if a positive number indicates a bank deposit, then a negative number indicates a withdrawal of the same amount. Negative numbers are usually written by writing a negative sign (also called a minus sign) in front of the number they are the opposite of. Thus the opposite of 7 is written −7. When the set of negative numbers is combined with the natural numbers and zero, the result is the set of integer numbers, also called integers, Z (German Zahl, plural Zahlen), also written \mathbb.### Rational numbers

A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero natural number denominator. The fraction m/n or- m \over n \,

- = \,.

### Real numbers

The real numbers
include all of the measuring numbers. Real numbers are usually
written using decimal
numerals, in which a decimal point is placed to the right of the
digit with place value one. 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. Thus

- 123.456\,

- -123.456\,.

Every rational number is also a real number. To
write a fraction as a decimal, divide the numerator by the
denominator. It is not the case, however, that every real number is
rational. If a real number cannot be written as a fraction of two
integers, it is called irrational.
A decimal that can be written as a fraction either ends
(terminates) or forever repeats, because it is the answer to a
problem in division. Thus the real number 0.5 can be written as 1/2
and the real number 0.333... (forever repeating threes) can be
written as 1/3. On the other hand, the real number π (pi), the ratio of the circumference of any
circle to its diameter,
is

- \pi = 3.14159265358979...\,.

- \sqrt = 1.41421356237 ...\,

Just as fractions can be written in more than one
way, so too can decimals. For example, if we multiply both sides of
the equation

- 1/3 = 0.333...\,

- 1 = 0.999...\,.

Every real number is either rational or
irrational. Every real number corresponds to a point on the
number
line. The real numbers also have an important but highly
technical property called the least
upper bound property. The symbol for the real numbers is R or
\mathbb.

When a real number represents a measurement, there is always
a margin of
error. This is often indicated by rounding or truncating a decimal, so that
digits that suggest a greater accuracy than the measurement itself
are removed. The remaining digits are called significant
digits. For example, measurements with a ruler can seldom be
made without a margin of error of at least 0.01 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 of 5.6088 square meters. Since only
the first two digits after the decimal place are significant, this
is usually rounded to 5.61. In abstract
algebra, the real numbers are up to isomorphism uniquely
characterized by being the only
complete ordered
field. They are not, however, an algebraically
closed field.

### 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 the question of whether a negative number can have a square root. This led to the invention of a new number: the square root of negative one, 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

In abstract
algebra, the complex numbers are an example of an algebraically
closed field, meaning that every polynomial with complex
coefficients can be
factored into
linear factors. Like the real number system, the complex number
system is a field
and is
complete, but unlike the real numbers it is not ordered. That
is, there is no meaning in saying that i is greater than 1, nor is
there any meaning in saying that that i is less than 1. In
technical terms, the complex numbers lack the trichotomy
property.

Complex numbers correspond to points on the
complex
plane, sometimes called the Argand plane.

Each of the number systems mentioned above is a
proper
subset of the next number system. Symbolically, N ⊂ Z ⊂ Q ⊂ R ⊂
C.

### Computable numbers

Moving to problems of computation, the computable
numbers are determined in the set of the real numbers. The
computable numbers, also known as the recursive numbers or the
computable reals, are the real numbers
that can be computed to within any desired precision by a finite,
terminating algorithm.
Equivalent definitions can be given using μ-recursive
functions, Turing
machines or λ-calculus
as the formal representation of algorithms. The computable numbers
form a real
closed field and can be used in the place of real numbers for
many, but not all, mathematical purposes.

### Other types

Hyperreal
and hypercomplex numbers are used in non-standard
analysis. The hyperreals, or nonstandard reals (usually denoted
as *R), denote an ordered
field which is a proper extension
of the ordered field of real numbers
R and which 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.

The idea behind p-adic
numbers is this: While real numbers may have infinitely long
expansions to the right of the decimal point, these numbers allow
for infinitely long expansions to the left. The number system which
results depends on what base is used for the digits: any
base is possible, but a system with the best mathematical
properties is obtained when the base is a prime
number.

For dealing with infinite collections, the
natural numbers have been generalized to the ordinal
numbers and to the cardinal
numbers. The former gives the ordering of the collection, while
the latter gives its size. For the finite set, the ordinal and
cardinal numbers are equivalent, but they differ in the infinite
case.

There are also other sets of numbers with
specialized uses. Some are subsets of the complex numbers. For
example, algebraic
numbers are the roots of polynomials with rational
coefficients.
Complex numbers that are not algebraic are called transcendental
numbers.

Sets of numbers that are not subsets of the
complex numbers are sometimes called hypercomplex
numbers. They include the quaternions H, invented by
Sir William
Rowan Hamilton, in which multiplication is not commutative, and the
octonions, in which
multiplication is not associative. Elements of
function fields of non-zero characteristic
behave in some ways like numbers and are often regarded as numbers
by number theorists.

In addition, various specific kinds of numbers
are studied in sets of natural
and integer
numbers.

An even number is an integer that is "evenly
divisible" by 2, i.e., divisible by 2 without remainder; an odd
number is an integer that is not evenly divisible by 2. (The
old-fashioned term "evenly divisible" is now almost always
shortened to "divisible".) A
formal definition of an odd number is that it is an integer of the
form n = 2k + 1, where k is an integer. An even number has the form
n = 2k where k is an integer.

A perfect number is defined as a
positive integer which is the sum of its proper positive
divisors, that is, the
sum of the positive divisors not including the number itself.
Equivalently, a perfect number is a number that is half the sum of
all of its positive divisors, or σ(n) = 2
n. The first perfect number is 6, because 1,
2, and 3 are its proper positive divisors and
1 + 2 + 3 = 6.
The next perfect number is 28 = 1 + 2 + 4 + 7 + 14.
The next perfect numbers are 496 and
8128 .
These first four perfect numbers were the only ones known to early
Greek
mathematics.

A figurate number is a number that can be
represented as a regular and discrete geometric pattern (e.g. dots).
If the pattern is polytopic, the figurate is
labeled a polytopic number, and may be a polygonal
number or a polyhedral number. Polytopic numbers for r = 2, 3,
and 4 are:

- P2(n) = 1/2 n(n + 1) (triangular numbers)
- P3(n) = 1/6 n(n + 1)(n + 2) (tetrahedral numbers)
- P4(n) = 1/24 n(n + 1)(n + 2)(n + 3) (pentatopic numbers)

## Numerals

Numbers should be distinguished from numerals,
the symbols used to represent numbers. The number five can be
represented by both the base ten numeral '5' and by the Roman
numeral 'V'. Notations used to represent numbers are discussed
in the article numeral
systems. An important development in the history of numerals
was the development of a positional system, like modern decimals,
which can represent very large numbers. The Roman numerals require
extra symbols for larger numbers.

## History

### History of integers

#### The first use of numbers

It is speculated that the first known use of numbers dates back to around 30000 BC, bones or other artifacts have been discovered with marks cut into them which are often considered tally marks. The use of these tally marks have been suggested to be anything from counting elapsed time, such as numbers of days, or keeping records of amounts.Tallying systems have no concept of place-value
(such as in the currently used decimal notation), which limit its
representation of large numbers and as such is often considered
that this is the first kind of abstract system that would be used,
and could be considered a 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.
http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html#berlin

#### History of zero

The use of zero as a number should be distinguished from its use as a placeholder numeral in place-value systems. Many ancient Indian texts use a Sanskrit word Shunya to refer to the concept of void; in mathematics texts this word would often be used to refer to the number zero. http://sunsite.utk.edu/math_archives/.http/hypermail/historia/apr99/0197.html. In a similar vein, Pāṇini (5th century BC) used the null (zero) operator (ie a lambda production) in the Ashtadhyayi, his algebraic grammar for the Sanskrit language. (also see Pingala)Records show that the Ancient
Greeks seemed unsure about the status of zero 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 zero and the vacuum.
The paradoxes
of Zeno of
Elea depend in large part on the uncertain interpretation of
zero. (The ancient Greeks even questioned if 1 was a
number.)

The late Olmec people of
south-central Mexico began to use
a true zero (a shell glyph) in the New World possibly by the
4th
century BC but certainly by 40 BC, which became
an integral part of Maya
numerals and the Maya
calendar, but did not influence Old World numeral
systems.

By 130, Ptolemy, influenced
by Hipparchus and
the Babylonians, was using a symbol for zero (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, nulla meaning nothing, not as a
symbol. When division produced zero as a remainder, nihil, 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.

An early documented use of the zero by Brahmagupta (in
the Brahmasphutasiddhanta)
dates to 628.
He treated zero as a number and discussed operations involving it,
including division.
By this time (7th century) the concept had clearly reached Cambodia, and
documentation shows the idea later spreading to China and the
Islamic
world.

#### History of negative numbers

The abstract concept of negative numbers was recognised as early as 100 BC - 50 BC. The Chinese ”Nine Chapters on the Mathematical Art” (Jiu-zhang Suanshu) contains methods for finding the areas of figures; red rods were used to denote positive coefficients, black for negative. This is the earliest known mention of negative numbers in the East; the first reference in a western work was in the 3rd century in Greece. Diophantus referred to the equation equivalent to 4x + 20 = 0 (the solution would be 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
Brahma-Sphuta-Siddhanta
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 debits (chapter 13 of Liber Abaci,
1202) and
later as losses (in Flos). At the same time, the Chinese were
indicating negative numbers by drawing a diagonal stroke through
the right-most nonzero digit of the corresponding positive number's
numeral. The first use of negative numbers in a European work was
by Chuquet
during the 15th
century. He used them as exponents, but referred to them
as “absurd numbers”.

As recently as the 18th
century, the Swiss
mathematician Leonhard
Euler believed that negative numbers were greater than infinity, and 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.

### History of rational, irrational, and real numbers

#### History of rational numbers

It is likely that the concept of fractional numbers dates to prehistoric times. Even the Ancient Egyptians wrote math texts describing how to convert general fractions into their special notation. 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 sutras to include calculations of
decimal-fraction approximations to pi or the square
root of two. Similarly, Babylonian math texts had always used
sexagesimal fractions with great frequency.

#### History of irrational numbers

The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800-500 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 his beliefs would not accept the existence of irrational numbers and so he sentenced Hippasus to death by drowning.The sixteenth century saw the final acceptance by
Europeans of
negative, integral and fractional
numbers. The seventeenth century saw decimal fractions with the
modern notation quite generally used by mathematicians. But it was
not until the nineteenth century that the irrationals were
separated into algebraic and transcendental parts, and a scientific
study of theory of irrationals was taken once more. It had remained
almost dormant since Euclid. The year
1872 saw the publication of the theories of Karl
Weierstrass (by his pupil Kossak), Heine
(Crelle,
74), Georg Cantor
(Annalen, 5), and Richard
Dedekind. Méray had taken
in 1869 the same point of departure as Heine, but
the theory is generally referred to the year 1872. Weierstrass's
method has been completely set forth by Salvatore
Pincherle (1880), and Dedekind's has received additional
prominence through the author's later work (1888) and the recent
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
(Crelle, 101), and Méray.

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 nineteenth 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
(1855) first connected the subject with determinants, resulting,
with the subsequent contributions of Heine,
Möbius, and Günther, in
the theory of Kettenbruchdeterminanten. Dirichlet also added to the
general theory, as have numerous contributors to the applications
of the subject.

#### Transcendental numbers and reals

The first results concerning transcendental numbers were Lambert's 1761 proof that π cannot be rational, and also that en is irrational if n is rational (unless n = 0). (The constant e was first referred to in Napier's 1618 work on logarithms.) Legendre extended this proof to showed that π is not the square root of a rational number. 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 (formula 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.Even the set of algebraic numbers was not
sufficient and the full set of real number includes transcendental
numbers. The existence of which 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 shows
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

The earliest known conception of mathematical infinity appears in the Yajur Veda, 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 circa 400 BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.In the West, the traditional notion of
mathematical infinity was defined by Aristotle, who
distinguished between actual
infinity and potential
infinity; the general consensus being that only the latter had
true value. Galileo'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. This was the first mathematical model that
represented infinity by numbers and gave rules for operating with
these infinite numbers.

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 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 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 (see Niccolo Fontana Tartaglia, 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.
The term "imaginary" for these quantities was coined by René
Descartes in 1637 and was meant to
be derogatory (see imaginary
number for a discussion of the "reality" of complex numbers). A
further source of confusion was that the equation

- \left ( \sqrt\right )^2 =\sqrt\sqrt=-1

- \sqrt\sqrt=\sqrt,

- \frac=\sqrt

The 18th century
saw the labors of Abraham
de Moivre and Leonhard
Euler. To De Moivre is due (1730) the well-known formula which
bears his name, de
Moivre's formula:

- (\cos \theta + i\sin \theta)^ = \cos n \theta + i\sin n \theta \,

and to Euler (1748) Euler's
formula of complex
analysis:

- \cos \theta + i\sin \theta = e ^. \,

The existence of complex numbers was not
completely accepted until the geometrical interpretation had been
described by Caspar
Wessel in 1799; it was
rediscovered several years later and popularized by Carl
Friedrich Gauss, 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 not a
little 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 a + bi, where a and b are integral, or
rational (and i is one of the two roots of x2 + 1 = 0). His
student, Ferdinand
Eisenstein, studied the type a + bω, where ω is a complex root
of x3 − 1 = 0. Other such classes (called cyclotomic
fields) of complex numbers are derived from the roots of
unity xk − 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. The general theory of fields was created by
Évariste
Galois, who studied the fields generated by the roots of any
polynomial equation F(x) = 0.

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 would eventually lead 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. The conjectures
of Goldbach and Riemann yet remain to be proved or refuted.

## References

- Erich Friedman, What's special about this number?
- 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.
- Whitehead and Russell, Principia Mathematica to *56, Cambridge University Press, 1910.
- What's a Number? at cut-the-knot

## See also

- Hebrew numerals
- Arabic numeral system
- Even and odd numbers
- Floating point representation in computers
- Large numbers
- List of numbers
- List of numbers in various languages
- Mathematical constants
- Mythical numbers
- Negative and non-negative numbers
- Orders of magnitude
- Physical constants
- Prime numbers
- Small numbers
- Subitizing and counting
- Number sign
- Numero sign
- Zero
- Pi
- The Foundations of Arithmetic

## External links

- Mesopotamian and Germanic numbers
- 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).

number in Arabic: عدد

number in Asturian: Númberu

number in Azerbaijani: Ədəd

number in Bengali: সংখ্যা

number in Min Nan: Sò͘-ba̍k

number in Belarusian (Tarashkevitsa): Лік

number in Breton: Niver

number in Bulgarian: Число

number in Catalan: Nombre

number in Czech: Číslo

number in Danish: Tal

number in German: Zahl

number in Estonian: Arv

number in Modern Greek (1453-): Αριθμός

number in Spanish: Número

number in Esperanto: Nombro

number in Basque: Zenbaki

number in Persian: عدد

number in Faroese: Tal

number in French: Nombre

number in Western Frisian: Getal

number in Scottish Gaelic: Àireamh

number in Galician: Número

number in Korean: 수 (수학)

number in Hindi: अंक

number in Croatian: Broj

number in Ido: Nombro

number in Indonesian: Bilangan

number in Interlingua (International Auxiliary
Language Association): Numero

number in Icelandic: Tala

number in Italian: Numero

number in Hebrew: מספר

number in Kannada: ಸಂಖ್ಯೆ

number in Georgian: რიცხვი

number in Haitian: Nonm

number in Kurdish: Hejmar

number in Latin: Numerus

number in Latvian: Skaitlis

number in Lithuanian: Skaičius

number in Lingala: Motángo

number in Hungarian: Szám

number in Macedonian: Број

number in Malagasy: Isa

number in Malayalam: സംഖ്യ

number in Malay (macrolanguage): Nombor

number in Burmese: နံပါတ်

nah:Tlapōhualli
number in Dutch: Getal

number in Japanese: 数

number in Norwegian: Tall

number in Norwegian Nynorsk: Tal

number in Narom: Neunmétho

number in Novial: Nombre

number in Occitan (post 1500): Nombre

number in Polish: Liczba

number in Portuguese: Número

number in Romanian: Număr

number in Quechua: Yupay

number in Russian: Число

number in Sicilian: Nùmmuru

number in Simple English: Number

number in Slovak: Číslo (matematika)

number in Slovenian: Število

number in Serbian: Број

number in Serbo-Croatian: Broj

number in Sundanese: Wilangan

number in Finnish: Luku

number in Swedish: Tal (matematik)

number in Tamil: எண்

number in Kabyle: Amḍan

number in Telugu: సంఖ్య

number in Thai: จำนวน

number in Vietnamese: Số (toán học)

number in Turkish: Sayı

number in Ukrainian: Число

number in Yiddish: צאל

number in Yoruba: Nọ́mbà

number in Chinese: 数 (数学)

# Synonyms, Antonyms and Related Words

G, M, a certain number, a few, a number,
account, act, add up, add up to, afterpiece, aggregate, aggregate to,
amount, amount to,
apportion, army, art, article, back matter, back
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over, call the roll, calling, career, career building, careerism, cast, census, chapter, character, chaser, chiliad, chiliagon, chiliahedron, chiliarch, chiliarchia, chunk, cipher, clan, clause, clutch, collection, color, come, come to, company, comprise, compute, contain, copy, count, covey, craft, crowd, curtain, curtain call, curtain
raiser, deal, decrease, denomination, description, designation, detail, difference, digit, divertimento, divertissement, divide, dose, dual, edition, enumerate, epilogue, exode, exodus, expository scene,
fascicle, feather, few, figure, figure up, finale, fix, foliate, folio, form, front matter, game, gang, gathering, genre, genus, gob, grain, grand, group, handful, handicraft, heap, hoke act, horde, host, hundred, hunk, ilk, impression, include, increase, integer, interlude, intermezzo, intermission, introduction, issue, itemize, kidney, kilo, kilocycle, kilogram, kilohertz, kiloliter, kilometer, kin, kind, label, lakh, large amount, legions, library, library edition,
lifework, lilt, line, line of business, line of
work, loads, lot, make, manner, many, mark, mass, measure, mess, meter, metier, metrics, millennium, millepede, milligram, milliliter, million, mission, mob, mold, mount up to, movement, multitude, myriad, mystery, nature, numbers, numeral, numerate, occupation, one hundred
thousand, one or two, outline, pack, page, paginate, paragraph, parcel, parse, part, party, passage, passel, persuasion, phrase, phylum, platoon, plural, poll, portion, practice, printing, product, profession, prologue, prosody, pursuit, quantify, quantity, quantize, race, racket, rate, ration, reckon, reckon up to, reckoning, reduce, resolve, rhythm, rhythmic pattern,
routine, run into, run
over, run to, scan,
scene, schematize, school edition,
score, section, sentence, series, set, several, shape, sheet, shtick, signature, singular, sketch, skit, slew, slews, small amount, song and
dance, sort, specialization, specialty, species, stamp, stand-up comedy act,
strain, stripe, striptease, style, sum, sum up, summation, swarm, swing, tale, tally, tell, ten thousand, text, the bottom line, the like of,
the likes of, the story, the whole story, thou, thousand, tons, tot up, tot up to, total, tote up to, trade, trade book, trade edition,
trial, tribe, troop, turn, two or three, type, umpteen, unitize, variety, verse, vocation, volume, walk, walk of life, whole, whole number, work, x number, yard