# Absolute value

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In mathematics, the absolute value (or modulus) |a| of a real number a is a's numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3.

Generalizations 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.

## Contents

### Terminology and notation

Jean-Robert Argand introduced the term "module" 'unit of measure' in French in 1806 specifically for the complex absolute value[1][2] and it was borrowed into English in 1866 as the Latin equivalent "modulus".[1] The term "absolute value" has been used in this sense since at least 1806 in French[3] and 1857 in English.[4] The notation | a | was introduced by Karl Weierstrass in 1841.[5] Other names for absolute value include "the numerical value"[1] and "the magnitude".[1]