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Absolute value
In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign. Namely, for a positive , for a negative (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.
Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.
==Terminology and notation==
In 1806, Jean-Robert Argand introduced the term ''module'', meaning ''unit of measure'' in French, specifically for the ''complex'' absolute value,〔Oxford English Dictionary, Draft Revision, June 2008〕〔(Nahin ), (O'Connor and Robertson ), and (functions.Wolfram.com. ); for the French sense, see Littré, 1877〕 and it was borrowed into English in 1866 as the Latin equivalent ''modulus''.〔 The term ''absolute value'' has been used in this sense from at least 1806 in French〔Lazare Nicolas M. Carnot, ''Mémoire sur la relation qui existe entre les distances respectives de cinq point quelconques pris dans l'espace'', p. 105 (at Google Books )〕 and 1857 in English.〔James Mill Peirce, ''A Text-book of Analytic Geometry'' (at Google Books ). The oldest citation in the 2nd edition of the Oxford English Dictionary is from 1907. The term ''absolute value'' is also used in contrast to ''relative value''.〕 The notation , with a vertical bar on each side, was introduced by Karl Weierstrass in 1841.〔Nicholas J. Higham, ''Handbook of writing for the mathematical sciences'', SIAM. ISBN 0-89871-420-6, p. 25〕 Other names for ''absolute value'' include ''numerical value''〔 and ''magnitude''.〔
The same notation is used with sets to denote cardinality; the meaning depends on context.
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