Absolute value

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

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.

1 Terminology and notation
2 Definition and properties
- 2.1 Real numbers
- 2.2 Complex numbers
3 Absolute value functions
- 3.1 Derivatives
4 Distance
5 Generalizations
6 Algorithms
7 See also
8 Notes
9 References
10 External links

Terminology and notation

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]

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. 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 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.^[6]

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.

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

Both the real and complex functions are idempotent.

It is a nonlinear function.

Derivatives

The derivative of the real absolute value function is the signum function, sgn(x), which is defined as

$\sgn (x) = \frac{x}{|x|},$

for x ≠ 0. 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. 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)abs(x).

The signum function is 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,

$u(x) = \frac{\sgn(x) +1}{2}.\,$

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.

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 = |x²|):

${\int|x|dx=x|x|-\int\frac{x^2}{|x|}dx= x|x|-\int|x|dx\iff 2\int|x|dx = x|x| \iff \int|x|dx =\frac{x|x|}{2}+C.}$

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 | a − b |, 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:^[7]

$d(a, b) \ge 0$	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(a − b), is a metric and the following are equivalent:

d satisfies the ultrametric inequality d(x, y) < max{d(x, z), d(y, z)}.

$\big\{ v\Big({\textstyle \sum_{k=1}^n } \mathbf{1}\Big)�: n \in \mathbb{N} \big\}$ 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.^[8]

Vector spaces

Main article: Norm (mathematics)

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.

Algorithms

Assembly language

Using x86 architecture assembly language, it is possible to take the absolute value of a register in just three instructions (example shown for a 32-bit register, Intel syntax):

cdq
xor eax, edx
sub eax, edx

cdq extends the sign bit of eax into edx. If eax is nonnegative, then edx becomes zero, and the latter two instructions have no effect, leaving eax unchanged. If eax is negative, then edx becomes 0xFFFFFFFF, or −1. The next two instructions then become a two's complement inversion, giving the absolute value of the negative value in eax. Note that the smallest negative value (−2³¹ or 0x80000000), which has no corresponding positive encoding, returns itself, which is accurate when taken as an unsigned integer.

C

In the C programming language, the abs, labs, llabs (in C99), fabs, fabsf, and fabsl functions, declared in math.h, compute the absolute value of an operand. Coding the integer version of the function is trivial, ignoring the boundary case where the smallest negative integer is input, this example uses the ternary operator:

int abs (int i) {
   return i < 0 ? -i : i;
}

The floating-point versions are trickier, as they have to contend with special codes for infinities and not-a-number.

Python

Python has a built-in function under the name of abs() that returns the absolute value of a number,^[9] the argument to the function may be an integer in which case the function returns an integer or a float in which case the function returns a float:

>>> abs(50)
50
>>> abs(-2)
2
>>> abs(-45.5)
45.5

The magnitude is returned if the argument is a complex number:^[9]

>>> abs(-3 + 4j)
5.0

Another function that can be used to calculate the absolute value is fabs() found in the math module, that can be accessed with import math. The difference between abs() and fabs() is that fabs() always returns a float:

>>> import math
>>> math.fabs(5)
5.0
>>> math.fabs(-366)
366.0
>>> math.fabs(-3.5)
3.5

A simple function to calculate the absolute value of an number by using the ternary operation and lambda:

>>> absolute_value = lambda number: number if number > 0 else -number
>>> absolute_value(2)
2
>>> absolute_value(-75)
75
>>> absolute_value(-5.63)
5.63

Notes

↑ ^1.0 ^1.1 ^1.2 ^1.3 Oxford English Dictionary, Draft Revision, June 2008
↑ Nahin, O'Connor and Robertson, and functions.Wolfram.com.; for the French sense, see Littré, 1877
↑ Lazare Nicolas M. Carnot, Mémoire sur la relation qui existe entre les distances respectives de cinq point quelconques pris dans l'espace, p. 105 at Google Books
↑ James Mill Peirce, A Text-book of Analytic Geometry at Google Books. The oldest citation in the 2nd edition of the Oxford English Dictionary is from 1907. The term "absolute value" is also used in contrast to "relative value".
↑ Nicholas J. Higham, Handbook of writing for the mathematical sciences, SIAM. ISBN 0898714206, p. 25
↑ Stewart, James B. (2001). Calculus: concepts and contexts. Australia: Brooks/Cole. ISBN 0-534-37718-1. , p. A5
↑ These axioms are not minimal; for instance, non-negativity can be derived from the other three: 0 = d(a, a) ≤ d(a, b) + d(b, a) = 2d(a, b).
↑ Schechter, p 260-261.
↑ ^9.0 ^9.1 Python's built-in functions

References

Nahin, Paul J.; An Imaginary Tale; Princeton University Press; (hardcover, 1998). ISBN 0-691-02795-1
O'Connor, J.J. and Robertson, E.F.; "Jean Robert Argand"
Schechter, Eric; Handbook of Analysis and Its Foundations, pp 259–263, "Absolute Values", Academic Press (1997) ISBN 0-12-622760-8

External links

absolute value at PlanetMath.
Weisstein, Eric W., "Absolute Value" from MathWorld.

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