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# Number

A number is an abstract entity used to describe quantity. There are different types of numbers. The most familiar numbers are the whole numbers {0, 1, 2, ...} denoted by W and the natural numbers {1, 2, 3, ...} used for counting and denoted by N. If the negative whole numbers are included, one obtains the integers Z. Ratios of integers are called rational numbers or fractions; the set of all rational numbers is denoted by Q. If all infinite and non-repeating decimal expansions are included, one obtains the real numbers R. Those real numbers which are not rational are called irrational numbers. Roots of polynomials with rational coefficients lead to algebraic 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:

$\mathbb{N}\sub\mathbb{W}\sub\mathbb{Z}\sub\mathbb{Q}\sub\mathbb{R}\sub\mathbb{C}$

Complex numbers can, in turn, be extended to quaternions, but multiplication of quaternions is not commutative. Octonions, in turn, extend the quaternions, but this time, associativity is lost. In fact, the only finite-dimensional associative division algebras over R are the reals, the complex numbers, and the quaternions. Elements of algebraic function fields of finite characteristic behave in many ways like numbers and are often regarded as a kind of number by number theorists.

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.

Many languages have the concept of grammatical number, an attribute of certain words and phrases that affects their syntactic usage and meaning.

## 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.