Number

A number is an abstract entity used to describe quantity. There are different 
types of numbers: the relationships between natural numbers, integers, rational 
numbers, irrational numbers and real numbers can be represented by a Venn 
diagram (in which the set of real numbers is viewed as a rectangle; with 
irrational numbers forming one part of the rectangle and rational numbers 
forming the other; and with integers being represented by a circle within 
the set of rational numbers, and natural numbers represented by a circle 
within the set of integers).

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.

Extensions

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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