A number is an abstract entity used to describe quantity. There are different types of numbers. The most familiar numbers are the natural numbers {0, 1, 2, ...} used for counting and denoted by N. If the negative whole numbers are included, one obtains the integersZ. 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 numbersR. Those real numbers which are not rational are called irrational numbers. The real numbers are in turn extended to the complex numbersC in order to be able to solve all algebraic equations. The above symbols are often written in blackboard bold, thus:
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.
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.
Newer developments are the hyperreal numbers and the surreal numbers, which 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, 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.)
