:
This article is about the concept in mathematics. For a discussion of numbers in linguistics, see grammatical number. For other uses, see #Disambiguation.
A
number is an abstract
entity used originally to describe
quantity. At least since the invention of
complex numbers, this definition must be relaxed. Preserving the main ideas of "quantity" except for the
total order, one can define numbers as elements of any
integral domain.
Examples
The most familiar numbers are the
natural numbers {0, 1, 2, ...} or {1, 2, 3, ...}, used for
counting, and denoted by
N.
If the negative whole numbers are included, one obtains the
integers Z.
("Whole numbers" are sometimes denoted by
W, but it depends on the author if this means
positive, non-negative, or all integers.)
Fractions of integers are called
rational numbers. The set of all rational numbers is denoted by
Q.
Rational numbers having a finite
decimal representation are called decimal fractions or decimal numbers, sometimes denoted by
D.
The
real numbers R, can have an infinite and non-repeating decimal expansion. Real numbers which are not rational are called
irrational numbers.
The real numbers can be extended to the
complex numbers C, which leads to an
algebraically closed field in which every polynomial with complex coefficients can be completely factored.
The above symbols are often written in
blackboard bold, thus:
:
In another respect, roots of polynomials with rational coefficients lead to
algebraic numbers, and those complex numbers which are not algebraic are called
transcendental numbers.
Iterating the adjunction of
imaginary units allows to extend
complex numbers to
quaternions H, losing commutativity of multiplication, and then to
octonions, losing
associativity and thus leaving the category of associative
division algebras.
Further generalizations
Elements of
function fields of finite
characteristic behave in some ways like numbers and are often regarded as a kind of number by number theorists.
Numerals and numbering
Numbers should be distinguished from
numerals, which are (combinations of)
symbols used to represent numbers. The notation of numbers as a series of digits is discussed in
numeral systems.
People like to assign numbers to objects in order to have unique names. There are various
numbering schemes for doing so.
Extensions
Superreal,
hyperreal and
surreal numbers extend the real numbers by adding infinitesimal and infinitely large numbers. While (most) real numbers have infinitely long expansions to the right of the decimal point, one can also try to allow for infinitely long expansions to the left in base p, where p is a
prime, leading to the
p-adic numbers. For dealing with infinite collections, the natural numbers have been generalized to the
ordinal numbers and to the
cardinal numbers. The former give the ordering of the collection, the latter its size. (For the finite case, the ordinal and cardinal numbers are equivalent; they diverge in the infinite case.)
The arithmetical operations of numbers, such as
addition,
subtraction,
multiplication and
division, are generalized in the branch of
mathematics called
abstract algebra; one obtains the
groups, rings and
fields.
See also
External links
Disambiguation
Category:Group theory
Category:Numbers
bg:Число
be:Лік
bn:সংখ্যা
br:Niver
ca:Nombre
cs:Číslo
da:Tal
de:Zahl
et:Arv
es:Número
eo:Nombro
eu:Zenbaki
fr:Nombre
gl:Número
fi:luku
hi:अंक
hr:Broj
id:Angka
is:Tala
it:Numero
he:מספר
hu:Szám
lt:Skaičius
nl:Getal
ja:数
no:Tall
pl:Liczba
pt:Número
ro:Număr
ru:Число
simple:Number
sl:Število
su:Wilangan
sv:Tal
th:จำนวน
tr:Sayı
zh:数 (数学)