Absolute value

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. Other names for absolute value include numerical value and magnitude. In programming languages and computational software packages, the absolute value of x is generally represented by abs(x), or a similar expression.

The vertical bar notation also appears in a number of other mathematical contexts: for example, when applied to a set, it is used to denote its cardinality; when applied to a matrix, it is used to denote its determinant. Thus, care must be taken to interpret vertical bars as an absolute value sign only when the argument is an algebraic object for which the notion of an absolute value is defined (e.g., an element of a normed division algebra like a real number, complex number, quaternion, etc). A closely related but distinct notation is the use of vertical bars for either the euclidean norm or sup norm of a vector in R n , although double vertical bars with subscripts ( | | ⋅ | | 2 and | | ⋅ | | ∞ , respectively) are a more common and less ambiguous notation.

Real numbers

For any real number x the absolute value or modulus of x is denoted by |x| (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 x is always either positive or zero, but never negative.

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 generalisation of the absolute value of the difference (see "Distance" below).

Since the square root notation without sign represents the positive square root, it follows that

for real values of a (equation (1)). This identity is sometimes used as a definition of absolute value of real numbers.

The absolute value has the following four fundamental properties (a, b are real numbers):

The properties given by equations (2)-(4) are readily apparent from the definition. To prove the triangle inequality (equation (5)), choose ε from { − 1 , 1 } so that ε ( a + b ) ≥ 0 . Since ε x ≤ | x | for real x regardless of the value of ε selected, (5) follows from the calculation | a + b | = ε ( a + b ) = ε a + ε b ≤ | a | + | b | . (For a generalization of this argument to complex numbers, see "Proof of the triangle inequality for complex numbers" below.)

Some additional useful properties are given below. Some of these are derived from and equivalent to the properties given by equations (2)-(5).

Two other useful properties concerning inequalities are:

These relations may be used to solve inequalities involving absolute values. For example:

Absolute value is used to define the absolute difference, the standard metric on the real numbers.

Complex numbers

Since the complex numbers are not ordered, the definition given above for the real absolute value cannot be directly generalised for a complex number. However the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalised. The absolute value of a complex number is defined as itsdistance in the complex plane from the origin using the Pythagorean theorem. More generally the absolute value of the difference of two complex numbers is equal to the distance between those two complex numbers.

In the context of abstract algebra, since the positive real numbers form a subgroup of the complex numbers under multiplication, we may think of absolute value as an endomorphism of the multiplicative group of the complex numbers.

For any complex number

where x and y are real numbers, the absolute value or modulus of z is denoted |z| and is given by

where Re(z) = x and Im(z) = y denote the real and imaginary parts of z, respectively. When the imaginary part y is zero this coincides with the definition of the absolute value of the real number x.

When a complex number z is expressed in polar form as

with r ≥ 0 and θ real, its absolute value is

The absolute value of a complex number can be written in the complex analogue of equation (1) above as:

where z is the complex conjugate of z. Note that, equation (1) is not, in general, true for complex z:

The complex absolute value shares all the properties of the real absolute value given in equations (2)–(11) above.

Importantly, the property of subadditivity ("triangle inequality", equation (5)) extends to any finite collection of complex numbers ( z k ) k = 1 n :

This inequality also applies to infinite series ( z k ) k ∈ N , provided ∑ k = 1 ∞ z k is absolutely convergent. If (Lebesgue) integration is viewed as the continuous analog of summation, then this inequality is analogously obeyed by complex-valued, measurable functions f : R → C when integrated over a measurable subset E :

(This includes Riemann-integrable functions over a bounded interval [ a , b ] as a special case.)

Proof of the triangle inequality for complex numbers

The triangle inequality, as given by ( ∗ ) , can be demonstrated by applying a few easily verified properties of the complex numbers: Namely, for every complex number z ∈ C , (i): there exists ω ∈ C such that | ω | = 1 , | z | = ω z , and (ii): R e ( z ) ≤ | z | ; and also, if ( w k ) k = 1 ℓ are complex numbers but ∑ k = 1 ℓ w k is real, then ∑ k = 1 ℓ w k = ∑ k = 1 ℓ R e ( w k ) .

Proof of ( ∗ ) : Choose c such that | c | = 1 , | ∑ k z k | = c ( ∑ k z k ) (summed over k = 1, 2, ... , n). The following computation then affords the desired inequality:

It is clear from the proof that the triangle inequality becomes an equality if and only if the c z k are non-negative real numbers for k = 1 , 2 , … , n , which in turn occurs if and only if all nonzero z k have the same argument (i.e., z k = a k ζ for a complex constant ζ and real constants a 1 , a 2 , … , a n ≥ 0 ).

After arguing that | f | is also measurable, the proof of the inequality ( ∗ ∗ ) proceeds via the same technique, by replacing ∑ k with ∫ E [ ⋅ ]   d x and z k with f ( x ) .

Absolute value function

The real absolute value function is continuous everywhere. It is differentiable everywhere except for x = 0. It is monotonically decreasing on the interval (−∞,0] and monotonically increasing on the interval [0,+∞). Since a real number and its opposite have the same absolute value, it is an even function, and is hence not invertible. The real absolute value function is a piecewise linear, convex function.

Both the real and complex functions are idempotent.

Relationship to the sign function

The absolute value function of a real number returns its value irrespective of its sign, whereas the sign (or signum) function returns a number's sign irrespective of its value. The following equations show the relationship between these two functions:

or

and for x ≠ 0,

Derivative

The real absolute value function has a derivative for every x ≠ 0, but is not differentiable at x = 0. Its derivative for x ≠ 0 is given by the step function:

The subdifferential of |x| at x = 0 is the interval [−1,1].

The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann equations.

The second derivative of |x| with respect to x is zero everywhere except zero, where it does not exist. As a generalised function, the second derivative may be taken as two times the Dirac delta function.

Antiderivative

The antiderivative (indefinite integral) of the absolute value function is

where C is an arbitrary constant of integration.

Distance

The absolute value is closely related to the idea of distance. As noted above, the absolute value of a real or complex number is the distance from that number to the origin, along the real number line, for real numbers, or in the complex plane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them.

The standard Euclidean distance between two points

and

in Euclidean n-space is defined as:

This can be seen to be a generalisation of |a − b|, since if a and b are real, then by equation (1),

While if

and

are complex numbers, then

The above shows that the "absolute value" distance for the real numbers or the complex numbers, agrees with the standard Euclidean distance they inherit as a result of considering them as the 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:

Ordered rings

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:

where −a is the additive inverse of a, and 0 is the additive identity element.

Fields

The fundamental properties of the absolute value for real numbers given in (2)–(5) above, 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) if it satisfies the following four axioms:

Where 0 denotes the additive identity element of F. It follows from positive-definiteness and multiplicativity that v(1) = 1, where 1 denotes the multiplicative identity element 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.

Vector spaces

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.

In the case of Euclidean space Rⁿ, the function defined by

is a norm called the Euclidean norm. When the real numbers R are considered as the one-dimensional vector space R¹, the absolute value is a norm, and is the p-norm (see L^p space) for any p. In fact the absolute value is the "only" norm on R¹, in the sense that, for every norm ‖·‖ on R¹, ‖x‖ = ‖1‖ ⋅ |x|. The complex absolute value is a special case of the norm in an inner product space. It is identical to the Euclidean norm, if the complex plane is identified with the Euclidean plane R².

Composition algebras

Every composition algebra A has an involution x → x* called its conjugation. The product in A of an element x and its conjugate x* is written N(x) = x x* and called the norm of x.

The real numbers ℝ, complex numbers ℂ, and quaternions ℍ are all composition algebras with norms given by definite quadratic forms. The absolute value in these division algebras is given by the square root of the composition algebra norm.

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).