From Academic Kids
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.
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.)
The above symbols are often written in blackboard bold, thus:
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.
Numerals and numbering
People like to assign numbers to objects in order to have unique names. There are various numbering schemes for doing so.
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.
- 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