Absolute value

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

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.

In 1806, Jean-Robert Argand introduced the term module, meaning unit of measure in French, 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 from at least 1806 in French[3] and 1857 in English.

[4] The notation |x|, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841.

[1] In programming languages and computational software packages, the absolute value of

The vertical bar notation also appears in a number of other mathematical contexts: for example, when applied to a set, it denotes its cardinality; when applied to a matrix, it denotes its determinant.

Vertical bars denote the absolute value only for algebraic objects for which the notion of an absolute value is defined, notably an element of a normed division algebra, for example a real number, a complex number, or a quaternion.

For example: The absolute value, as "distance from zero", is used to define the absolute difference between arbitrary real numbers, the standard metric on the real numbers.

However, the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalised.

is zero, this coincides with the definition of the absolute value of the real number

Since a real number and its opposite have the same absolute value, it is an even function, and is hence not invertible.

, then the following relationship to the minimum and maximum functions hold: and The formulas can be derived by considering each case

The real absolute value function has a derivative for every x ≠ 0, but is not differentiable at x = 0.

The absolute value is closely related to the idea of distance.

complex numbers, i.e. in a 2-space, The above shows that the "absolute value"-distance, for real and complex numbers, agrees with the standard Euclidean distance, which they inherit as a result of considering them as one and two-dimensional Euclidean spaces, respectively.

The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion of a distance function as follows: A real valued function d on a set X × X is called a metric (or a distance function) on X, if it satisfies the following four axioms:[15] The definition of absolute value given for real numbers above can be extended to any ordered ring.

That is, if a is an element of an ordered ring R, then the absolute value of a, denoted by |a|, is defined to be:[16] where −a is the additive inverse of a, 0 is the additive identity, and < and ≥ have the usual meaning with respect to the ordering in the ring.

The four fundamental properties of the absolute value for real numbers can be used to generalise the notion of absolute value to an arbitrary field, as follows.

A real-valued function v on a field F is called an absolute value (also a modulus, magnitude, value, or valuation)[17] if it satisfies the following four axioms: Where 0 denotes the additive identity of F. It follows from positive-definiteness and multiplicativity that v(1) = 1, where 1 denotes the multiplicative identity of F. The real and complex absolute values defined above are examples of absolute values for an arbitrary field.

If v is an absolute value on F, then the function d on F × F, defined by d(a, b) = v(a − b), is a metric and the following are equivalent: An absolute value which satisfies any (hence all) of the above conditions is said to be non-Archimedean, otherwise it is said to be Archimedean.

[18] Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, to generalise the notion to an arbitrary vector space.

A real-valued function on a vector space V over a field F, represented as ‖ · ‖, is called an absolute value, but more usually a norm, if it satisfies the following axioms: For all a in F, and v, u in V, The norm of a vector is also called its length or magnitude.

The complex absolute value is a special case of the norm in an inner product space, which is identical to the Euclidean norm when the complex plane is identified as the Euclidean plane

The product in A of an element x and its conjugate x* is written N(x) = x x* and called the norm of x.

are all composition algebras with norms given by definite quadratic forms.

In general the norm of a composition algebra may be a quadratic form that is not definite and has null vectors.

However, as in the case of division algebras, when an element x has a non-zero norm, then x has a multiplicative inverse given by x*/N(x).