Absolute value

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Absolute value

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.  The graph of the absolute value function for real numbers.

Terminology and notation

Jean-Robert Argand introduced the term "module" 'unit of measure' in French in 1806 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 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".

The same notation is used with sets to denote cardinality; the meaning depends on context.

Definition and properties

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 $|a| = \begin{cases} a, & \mbox{if } a \ge 0 \\ -a, & \mbox{if } a < 0. \end{cases}$

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 $|a| = \sqrt{a^2}$ (1)

which is sometimes used as a definition of absolute value.

The absolute value has the following four fundamental properties: $|a| \ge 0$ (2) Non-negativity $|a| = 0 \iff a = 0$ (3) Positive-definiteness $|ab| = |a||b|\,$ (4) Multiplicativeness $|a+b| \le |a| + |b|$ (5) Subadditivity

Other important properties of the absolute value include: $|-a| = |a|\,$ (6) Symmetry $|a - b| = 0 \iff a = b$ (7) Identity of indiscernibles (equivalent to positive-definiteness) $|a - b| \le |a - c| +|c - b|$ (8) Triangle inequality (equivalent to subadditivity) $|a/b| = |a| / |b| \mbox{ (if } b \ne 0) \,$ (9) Preservation of division (equivalent to multiplicativeness) $|a-b| \ge ||a| - |b||$ (10) (equivalent to subadditivity)

If b > 0, two other useful properties concerning inequalities are: $|a| \le b \iff -b \le a \le b$ $|a| \ge b \iff a \le -b \mbox{ or } b \le a$

These relations may be used to solve inequalities involving absolute values. For example: $|x-3| \le 9$ $\iff -9 \le x-3 \le 9$ $\iff -6 \le x \le 12$

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

Complex numbers  The absolute value of a complex number z is the distance r from z to the origin. It is also seen in the picture that z and its complex conjugate z have the same absolute value.

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 identity given in equation (1) above: $|a| = \sqrt{a^2}$

can be seen as motivating the following definition.

For any complex number $z = x + iy,\,$

where x and y are real numbers, the absolute value or modulus of z is denoted |z| and is defined as $|z| = \sqrt{x^2 + y^2}.$

It follows that the absolute value of a real number x is equal to its absolute value considered as a complex number since: $|x + i0| = \sqrt{x^2 + 0^2} = \sqrt{x^2} = |x|.$

Similar to the geometric interpretation of the absolute value for real numbers, it follows from the Pythagorean theorem that the absolute value of a complex number is the distance in the complex plane of that complex number from the origin, and more generally, that the absolute value of the difference of two complex numbers is equal to the distance between those two complex numbers.

The complex absolute value shares all the properties of the real absolute value given in (2)–(10) above. In addition, If $z = x + i y = r (\cos \phi + i \sin \phi ) \,$

and $\overline{z} = x - iy$

is the complex conjugate of z, then it is easily seen that \begin{align} |z| & = r, \\ |z| & = |\overline{z}|\end{align}

and $|z| = \sqrt{z\overline{z}},$

with the last formula being the complex analogue of equation (1) mentioned above in the real case.

The absolute square of z is defined as $|z|^2 = z\overline{z} = x^2 + y^2.$

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.

Absolute value functions

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 function.

It is also a convex function.

Derivatives

The derivative of the real absolute value function is given by $\frac{d|x|}{dx} = \frac{x}{|x|} = \begin{cases} -1 & x<0 \\ 1 & x>0 \end{cases}$

for x ≠ 0 (cf. signum function). The absolute value function is not differentiable at x = 0. For applications in which a well-defined derivative may be needed, however, the subderivative is well defined at zero.

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

The absolute value function has no concavity at any point, the sign function is constant at all points. Therefore the second derivative of |x| with respect to x is zero everywhere except zero, where it is undefined.

Antiderivative

The absolute value function is also integrable. Its antiderivative is $\int|x|dx=\frac{x|x|}{2}+C,$

as evidenced by the following (using integration by parts and the fact that x2 = |x|2): ${\int|x|dx=x|x|-\int\frac{x^2}{|x|}dx= x|x|-\int|x|dx\iff 2\int|x|dx = x|x| \iff \int|x|dx =\frac{x|x|}{2}+C.}$

Relationship to other functions

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. Therefore x = sgn(x)|x|. The real absolute value function is also related to a form of the Heaviside step function used in signal processing, defined as: $u(x) = \begin{cases} 0, & x < 0 \\ \frac{1}{2}, & x = 0 \\ 1, & x > 0, \end{cases}$

where the value of the Heaviside function at zero is conventional. So for all nonzero points on the real number line, $u(x) = \frac{\sgn(x) +1}{2}.\,$

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 $a = (a_1, a_2, \dots , a_n)$

and $b = (b_1, b_2, \dots , b_n)$

in Euclidean n-space is defined as: $\sqrt{(a_1-b_1)^2 + (a_2-b_2)^2 + \cdots + (a_n-b_n)^2}.$

This can be seen to be a generalization of | ab |, since if a and b are real, then by equation (1), $|a - b| = \sqrt{(a - b)^2}.$

While if $a = a_1 + i a_2 \,$

and $b = b_1 + i b_2 \,$

are complex numbers, then $|a - b| \,$ $= |(a_1 + i a_2) - (b_1 + i b_2)|\,$ $= |(a_1 - b_1) + i(a_2 - b_2)|\,$ $= \sqrt{(a_1 - b_1)^2 + (a_2 - b_2)^2}.$

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: $d(a, b) \ge 0$ Non-negativity $d(a, b) = 0 \iff a = b$ Identity of indiscernibles $d(a, b) = d(b, a) \,$ Symmetry $d(a, b) \le d(a, c) + d(c, b)$ Triangle inequality

Generalizations

Ordered rings

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: $|a| = \begin{cases} a, & \mbox{if } a \ge 0 \\ -a, & \mbox{if } a < 0, \end{cases}$

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 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) if it satisfies the following four axioms: $v(a) \ge 0$ Non-negativity $v(a) = 0 \iff a = \mathbf{0}$ Positive-definiteness $v(ab) = v(a) v(b) \,$ Multiplicativeness $v(a+b) \le v(a) + v(b)$ 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(ab), is a metric and the following are equivalent:

• d satisfies the ultrametric inequality $d(x, y) \leq \max(d(x,z),d(y,z))$ for all x, y, z in F.
• $\big\{ v\Big({\textstyle \sum_{k=1}^n } \mathbf{1}\Big) : n \in \mathbb{N} \big\}$ is bounded in R.
• $v\Big({\textstyle \sum_{k=1}^n } \mathbf{1}\Big) \le 1 \text{ for every } n \in \mathbb{N}.$
• $v(a) \le 1 \Rightarrow v(1+a) \le 1 \text{ for all } a \in F.$
• $v(a + b) \le \mathrm{max}\{v(a), v(b)\} \text{ for all } a, b \in F.$

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 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, $\|\mathbf{v}\| \ge 0$ Non-negativity $\|\mathbf{v}\| = 0 \iff \mathbf{v} = 0$ Positive-definiteness $\|a \mathbf{v}\| = |a| \|\mathbf{v}\|$ Positive homogeneity or positive scalability $\|\mathbf{v} + \mathbf{u}\| \le \|\mathbf{v}\| + \|\mathbf{u}\|$ Subadditivity or the triangle inequality

The norm of a vector is also called its length or magnitude.

In the case of Euclidean space Rn, the function defined by $\|(x_1, x_2, \dots , x_n) \| = \sqrt{\sum_{i=1}^{n}(x_i)^2}$

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 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.

Smooth approximation

The absolute value function does not have a derivative at x=0[citation needed]. Sometimes, a smooth approximation is required that has to be smooth at x=0. One of such approximations is given by: $|x| \approx \frac{2x}{\pi} \mathrm{arctan}(kx)$

where k>0. The approximation improves as k increases. k=10 is good for many practical purposes.

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