Absolute value

In mathematics, the absolute value or modulus of a real number

x

{\displaystyle x}

, denoted

|

x

|

{\displaystyle |x|}

, is the (non-negative) magnitude of

x

{\displaystyle x}

measured without regard to its sign. Namely,

|

x

|

=

x

{\displaystyle |x|=x}

if

x

{\displaystyle x}

is a positive number, and

|

x

|

=

−

x

{\displaystyle |x|=-x}

if

x

{\displaystyle x}

is negative (in which case

−

x

{\displaystyle -x}

is positive), and

|

0

|

=

0

{\displaystyle |0|=0}

. 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, 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 and 1857 in English. The notation |x|, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841.