1.a real number regardless of its sign
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valeur algébrique (fr)[Classe]
arithmétique (fr)[termes liés]
absolute value (n.)↕
absolute value (n.)↕
amount, measure, quantity[Hyper.]
mathematics[Domaine]
AbsoluteValueFn[Domaine]
definite quantity[Hyper.]
absolute value (n.)↕
In mathematics, the absolute value (or modulus) | a | of a real number a is the numerical value of a without regard to its sign. So, 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.
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.
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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]}
The same notation is used with sets to denote cardinality; the meaning depends on context.
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^{[6]}
As can be seen from the above definition, the absolute value of a 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 generalization 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
( | )
which is sometimes used as a definition of absolute value.^{[7]}
The absolute value has the following four fundamental properties:
( | )Non-negativity | |
( | )Positive-definiteness | |
( | )Multiplicativeness | |
( | )Subadditivity |
Other important properties of the absolute value include:
( | )Self-composition (the absolute value of the absolute value is the absolute value) | |
( | )Symmetry | |
( | )Identity of indiscernibles (equivalent to positive-definiteness) | |
( | )Triangle inequality (equivalent to subadditivity) | |
( | )Preservation of division (equivalent to multiplicativeness) | |
( | )(equivalent to subadditivity) |
If b > 0, 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.
Since the complex numbers are not ordered, the definition given above for the real absolute value cannot be directly generalized for a complex number. However the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalized. The absolute value of a complex number is defined as its distance 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.
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
When the complex part y is zero this is the same as 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 is the complex conjugate of z.
The complex absolute value shares all the properties of the real absolute value given in equations (2)–(11) above.
Since the positive reals 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.^{[citation needed]}
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 negative have the same absolute value, it is an even function, and is hence not invertible.
Both the real and complex functions are idempotent.
It is a nonlinear convex function.
The derivative of the real absolute value function is given by the step function^{[8]}^{[9]}
The subdifferential of at is the interval [−1,1].^{[10]}
The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann equations.^{[8]}
The second derivative of | x | with respect to x is zero everywhere except zero, where it is undefined. As a generalized function, the second derivative may be taken to equal two times the Dirac delta function.
The antiderivative (indefinite integral) of the absolute value function is
where C is an arbitrary constant of integration.
Where the absolute value function of a real number returns a value without respect to its sign, the signum function returns a number's sign without respect to its value. The following equations show the relationship between these two functions:
The real absolute value function is also related to a form of the Heaviside step function used in signal processing, defined as:
where the value of the Heaviside function at zero is conventional. So for all nonzero points on the real number line,
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 generalization 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 distance function (or a metric) on X, if it satisfies the following four axioms:^{[11]}
Non-negativity | |
Identity of indiscernibles | |
Symmetry | |
Triangle inequality |
The definition of absolute value given for real numbers above can easily 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:^{[12]}
where −a is the additive inverse of a, and 0 is the additive identity element.
The fundamental properties of the absolute value for real numbers given in (2)–(5) above, can be used to generalize 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)^{[13]} if it satisfies the following four axioms:
Non-negativity | |
Positive-definiteness | |
Multiplicativeness | |
Subadditivity or the triangle inequality |
Where 0 denotes the additive identity element of F. It follows from positive-definiteness and multiplicativeness 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.^{[14]}
Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, to generalize the notion to an arbitrary vector space.
A real-valued function on a vector space V over a field F, represented as ||V||, is called an absolute value (or more usually a norm) if it satisfies the following axioms:
For all a in F, and v, u in V,
Non-negativity | |
Positive-definiteness | |
Positive homogeneity or positive scalability | |
Subadditivity or the triangle inequality |
The norm of a vector is also called its length or magnitude.
In the case of Euclidean space R^{n}, the function defined by
is a norm called the Euclidean norm. When the real numbers R are considered as the one-dimensional vector space R^{1}, the absolute value is a norm, and is the p-norm for any p. In fact the absolute value is the "only" norm on R^{1}, in the sense that, for every norm || · || on R^{1}, || 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 R^{2}.
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