Number
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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.
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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:
- <math>\mathbb{N}\sub\mathbb{Z}\sub\mathbb{D}\sub\mathbb{Q}\sub\mathbb{R}\sub\mathbb{C}<math>
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
- Arabic numeral system
- even and odd numbers
- floating point numbers are used by computers to approximate real numbers
- large numbers
- list of numbers
- mathematical constant
- mythical number
- negative and non-negative numbers
- orders of magnitude (numbers)
- physical constant
- prime number
- small number
- subitizing and counting
- numbers in various languages
- number sign
- numero sign