Number

A number is an abstract entity used to describe quantity. The most familiar numbers are the natural numbers 0, 1, 2, ... used for counting. If negative numbers are added, one obtains the integers. Ratios of integers are called rational numbers. If all infinite and non-repeating decimal expansions are included, one obtains the real numbers, which are in turn extended to the complex numbers in order to be able to solve all algebraic equations. Newer developments are the hyperreal numbers and the surreal numbers which extend the real numbers by adding infinitesimal and infinitely large numbers.

Instead of allowing arbitrary infinitely long expansions to the right of the decimal point, which leads from the rational to the real numbers, one can also try to allow for infinitely long expansions to the left of the decimal point, leading to p-adic numbers.

For measuring the size of infinite sets, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers.

The arithmetical operations of numbers, such as addition and multiplication, are generalized in the branch of mathematics called abstract algebra; one obtains the algebraic structures group, ring and field.