for: value (ethics)

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

The absolute value of a number $a$ is denoted by $|a|$.

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.

Terminology and notation

Jean-Robert Argand introduced the term "module" 'unit of measure' in French in 1806 specifically for the complex absolute valueOxford English Dictionary, Draft Revision, June 2008 and it was borrowed into English in 1866 as the Latin equivalent "modulus".

The term "absolute value" has been used in this sense since at least 1806 in French and 1857 in English.

The notation | a | was introduced by Karl Weierstrass in 1841.

Other names for absolute value include "the numerical value" and "the magnitude", that is, ignoring the sign.

Real numbers

For any real number a the absolute value or modulus of a is denoted by | a | (a vertical bar on each side of the quantity) and is defined as

As can be seen from the above definition, the absolute value of a is always either positive or zero, but never negative. The same notation is used with sets to denote cardinality; the meaning depends on context.

From an analytic geometry point of view, the absolute value of a real number is that number's distance from zero along the real number line, and more generally the absolute value of the difference of two real numbers is the distance between them. Indeed the notion of an abstract distance function in mathematics can be seen to be a generalization of the absolute value of the difference (see "Distance" below).

Since the square-root function is normally defined as the positive square root, $|a|\; =\; \backslash sqrt\{a^2\}$, which is sometimes even used as a definition of absolute value.

The absolute value has the following four fundamental properties:

Other important properties of the absolute value include:

b \ne 0) \,

|Preservation of division (equivalent to multiplicativeness) |- |$|a-b|\; \backslash ge\; ||a|\; -\; |b||$ |(equivalent to subadditivity) |}

If b > 0, two other useful inequalities are:

$|a|\; \backslash le\; b\; \backslash iff\; -b\; \backslash le\; a\; \backslash le\; b$