Numbers should be distinguished from numerals which are symbols used to represent numbers. The notation of numbers as series of digits is discussed in numeral systems. People like to assign numbers to objects. There are various numbering schemes for doing so. The arithmetical operations of numbers, such as addition and multiplication, are generalized in the branch of mathematics called abstract algebra; one obtains the groups, rings and fields.
The most familiar numbers are the natural numbers (0, 1, 2, ...) used for counting[?] and denoted by N. If negative 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, which are in turn extended to the complex numbers C in order to be able to solve all algebraic equations. The above symbols are generally written in blackboard bold.
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 measuring the size of infinite sets, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers.
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