Absolute value

File:Absolute value.svg

The graph of the absolute value function for real numbers

File:AbsoluteValueDiagram.svg

The absolute value of a number may be thought of as its distance from zero.

In mathematics, the absolute value or modulus of a real number $x$, denoted $|x|$, is the non-negative value of $x$ without regard to its sign. Namely, $|x|=x$ if $x$ is a positive number, and $|x|=-x$ if $x$ is negative (in which case negating $x$ makes $-x$ positive), and $|0|=0$. 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.

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.

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 Template:Math, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841. Other names for absolute value include numerical value and magnitude. The absolute value of $x$ has also been denoted $\mathrm{abs}x$ in some mathematical publications,Template:Sfnp and in spreadsheets, programming languages, and computational software packages, the absolute value of $x$ is generally represented by abs(x), or a similar expression,Template:Sfnp as it has been since the earliest days of high-level programming languages.Template:Sfnp

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.Template:Sfnp 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. A closely related but distinct notation is the use of vertical bars for either the Euclidean normTemplate:Sfnp or sup normTemplate:Sfnp of a vector in ${\mathbb{ℝ}}^n$, although double vertical bars with subscripts ($\Vert \cdot {\Vert }_2$ and $\Vert \cdot {\Vert }_{\mathrm{\infty }}$, respectively) are a more common and less ambiguous notation.

Definition and properties

Real numbers

For any real number $x$, the absolute value or modulus of $x$ is denoted by $|x|$, with a vertical bar on each side of the quantity, and is defined asTemplate:Sfnp $$|x|=\{\begin{array}{ll}x,& \text{if }x\ge 0\\ -x,& \text{if }x<0.\end{array}$$

The absolute value of $x$ is thus always either a positive number or zero, but never negative. When $x$ itself is negative ($x<0$), then its absolute value is necessarily positive ($|x|=-x>0$).Template:Sfnp

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 (their absolute difference) is the distance between them.Template:Sfnp The notion of an abstract distance function in mathematics can be seen to be a generalisation of the absolute value of the difference.Template:Sfnp See Template:Slink below.

Since the square root symbol represents the unique positive square root, when applied to a positive number, it follows thatTemplate:Sfnp} This is equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers.Template:Sfnp

The absolute value has the following four fundamental properties ($a$, $b$ are real numbers), that are used for generalization of this notion to other domains:Template:Sfnp

	Non-negativityTemplate:Sfnp
$|a|=0⟺a=0$	Positive-definitenessTemplate:Sfnp
$|ab|=\left|a\right|\left|b\right|$	MultiplicativityTemplate:Sfnp
	Subadditivity, specifically the triangle inequalityTemplate:Sfnp

Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition. To see that subadditivity holds, first note that $|a+b|=s(a+b)$ where $s=\pm 1$, with its sign chosen to make the result positive. Now, since $-1\cdot x\le |x|$ and $+1\cdot x\le |x|$, it follows that, whichever of $\pm 1$ is the value of $s$, one has $s\cdot x\le |x|$ for all real $x$. Consequently, $|a+b|=s\cdot (a+b)=s\cdot a+s\cdot b\le |a|+|b|$, as desired.

Some additional useful properties are given below. These are either immediate consequences of the definition or implied by the four fundamental properties above.

$\left|\left|a\right|\right|=|a|$	Idempotence (the absolute value of the absolute value is the absolute value)
$|-a|=|a|$	Evenness (reflection symmetry of the graph)Template:Sfnp
$|a-b|=0⟺a=b$	Identity of indiscernibles (equivalent to positive-definiteness)
$|a-b|\le |a-c|+|c-b|$	Triangle inequality (equivalent to subadditivity)
$\left|\frac{a}{b}\right|=\frac{|a|}{|b|}$ (if $b\ne 0$)	Preservation of division – equivalent to multiplicativityTemplate:Sfnp
$|a-b|\ge |\left|a\right|-\left|b\right||$	Reverse triangle inequality – equivalent to subadditivityTemplate:Sfnp

Two other useful properties concerning inequalities are:Template:Sfnp

$|a|\le b⟺-b\le a\le b$

$|a|\ge b⟺a\le -b$ or $a\ge b$

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

$|x-3|\le 9$	$⟺-9\le x-3\le 9$
	$⟺-6\le x\le 12$

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.

Complex numbers

File:Complex conjugate picture.svg

The absolute value of a complex number $z$ is the distance $r$ of $z$ from the origin. It is also seen in the picture that $z$ and its complex conjugate $\overline{z}$ have the same absolute value.

Since the complex numbers are not ordered, the definition given at the top for the real absolute value cannot be directly applied to complex numbers. 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 by the Euclidean distance of its corresponding point in the complex plane from the origin. This can be computed using the Pythagorean theorem: 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 byTemplate:Sfnp $$|z|=\sqrt{\mathrm{Re}(z{)}^2+\mathrm{Im}(z{)}^2}=\sqrt{x^2+y^2},$$ the Pythagorean addition of $x$ and $y$, where $\mathrm{Re}(z)=x$ and $\mathrm{Im}(z)=y$ 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 $x$.Template:Sfnp

When a complex number $z$ is expressed in its polar form as $z=r{e}^{i\theta },$ its absolute value is $|z|=r.$

Since the product of any complex number $z$ and its complex conjugate $\overline{z}=x-iy$, with the same absolute value, is always the non-negative real number $(x^2+y^2)$, the absolute value of a complex number $z$ is the square root of $z\cdot \bar{z},$ which is therefore called the absolute square or squared modulus of $z$:Template:Sfnp $$|z|=\sqrt{z\cdot \bar{z}}.$$ This generalizes the alternative definition for reals: $|x|=\sqrt{x\cdot x}$.

The complex absolute value shares the four fundamental properties given above for the real absolute value. The identity $|z{|}^n=|z^n|$ is a special case of multiplicativity that is often useful by itself.Template:Sfnp

Absolute value function

File:Absolute value.svg

The graph of the absolute value function for real numbers

File:Absolute value composition.svg

Composition of absolute value with a cubic function in different orders

The real absolute value function is continuous everywhere. It is differentiable everywhere except for Template:Math. It is monotonically decreasing on the interval Template:Open-closed and monotonically increasing on the interval Template:Closed-open.Template:Sfnp Since a real number and its opposite have the same absolute value, it is an even function, and is hence not invertible.Template:Sfnp The real absolute value function is a piecewise linear, convex function.Template:Sfnp

For both real and complex numbers, the absolute value function is idempotent (meaning that the absolute value of any absolute value is itself).

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:

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

or

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

and for Template:Math,

$\mathrm{sgn}(x)=\frac{|x|}{x}=\frac{x}{|x|}.$

Relationship to the max and min functions

Let $s,t\in \mathbb{ℝ}$, then the following relationship to the minimum and maximum functions hold:

$|t-s|=-2\min (s,t)+s+t$

and

$|t-s|=2\max (s,t)-s-t.$

The formulas can be derived by considering each case $s>t$ and $t>s$ separately.

From the last formula one can derive also $|t|=\max (t,-t)$.

Derivative

The real absolute value function has a derivative for every Template:Math, given by a step function equal to the sign function except at Template:Math where the absolute value function is not differentiable:Template:Sfnp $$\begin{array}{rl}\frac{d\left|x\right|}{dx}& =\frac{x}{|x|}=\{\begin{array}{ll}-1& x<0\\ 1& x>0\end{array}\\ & =\mathrm{sgn}x\text{for }x\ne 0.\end{array}$$

The real absolute value function is an example of a continuous function that achieves a global minimum where the derivative does not exist.

The subdifferential of Template:Math at Template:Math is the interval Template:Closed-closed.Template:Sfnp

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

The second derivative of Template:Math with respect to Template:Mvar 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 real absolute value function is

$\int \left|x\right|dx=\frac{x\left|x\right|}{2}+C,$

where Template:Mvar is an arbitrary constant of integration. This is not a complex antiderivative because complex antiderivatives can only exist for complex-differentiable (holomorphic) functions, which the complex absolute value function is not.

Derivatives of compositions

The following two formulae are special cases of the chain rule:

$\frac{d}{dx}f(|x|)=\frac{x}{|x|}(f^{\prime }(|x|))$

if the absolute value is inside a function, and

$\frac{d}{dx}|f(x)|=\frac{f(x)}{|f(x)|}{f}^{\prime }(x)$

if another function is inside the absolute value. In the first case, the derivative is always discontinuous at $x=0$ in the first case and where $f(x)=0$ in the second case.

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 space|Euclidean Template:Mvar-space]] is defined as:Template:Sfnp $$\sqrt{{\sum }_{i=1}^n(a_i-b_i{)}^2}.$$

This can be seen as a generalisation, since for $a_1$ and $b_1$ real, i.e. in a 1-space, according to the alternative definition of the absolute value,

$|a_1-b_1|=\sqrt{(a_1-b_1{)}^2}=\sqrt{{\sum }_{i=1}^1(a_i-b_i{)}^2},$

and for $a=a_1+i{a}_2$ and $b=b_1+i{b}_2$ complex numbers, i.e. in a 2-space,

$|a-b|$	$=|(a_1+i{a}_2)-(b_1+i{b}_2)|$
	$=|(a_1-b_1)+i(a_2-b_2)|$
	$=\sqrt{(a_1-b_1{)}^2+(a_2-b_2{)}^2}=\sqrt{{\sum }_{i=1}^2(a_i-b_i{)}^2}.$

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 Template:Mvar on a set Template:Math is called a metric (or a distance function) on Template:Mvar, if it satisfies the following four axioms:

$d(a,b)\ge 0$	Non-negativity
$d(a,b)=0⟺a=b$	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 be extended to any ordered ring. That is, if Template:Mvar is an element of an ordered ring R, then the absolute value of Template:Mvar, denoted by Template:Math, is defined to be: $$|a|=\{\begin{array}{rl}a,& \text{if }a\ge 0\\ -a,& \text{if }a<0.\end{array}$$ where Template:Math is the additive inverse of Template:Mvar, 0 is the additive identity, and < and ≥ have the usual meaning with respect to the ordering in the ring.Template:Sfnp

Fields

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 Template:Mvar on a field Template:Mvar is called an absolute value (also a modulus, magnitude, value, or valuation)Template:SfnpTemplate:Efn if it satisfies the following four axioms:

$v(a)\ge 0$	Non-negativity
$v(a)=0⟺a=\mathbf{𝟎}$	Positive-definiteness
$v(ab)=v(a)v(b)$	Multiplicativity
$v(a+b)\le v(a)+v(b)$	Subadditivity or the triangle inequality

Where 0 denotes the additive identity of Template:Mvar. It follows from positive-definiteness and multiplicativity that Template:Math, where 1 denotes the multiplicative identity of Template:Mvar. The real and complex absolute values defined above are examples of absolute values for an arbitrary field.

If Template:Mvar is an absolute value on Template:Mvar, then the function Template:Mvar on Template:Math, defined by Template:Math, is a metric and the following are equivalent:

• Template:Mvar satisfies the ultrametric inequality $d(x,y)\le \max (d(x,z),d(y,z))$ for all Template:Mvar, Template:Mvar, Template:Mvar in Template:Mvar.
• $\{v\left({\sum }_{k=1}^n\mathbf{𝟏}\right):n\in \mathbb{\mathbb{N}}\}$ is bounded in R.
• $v\left({\sum }_{k=1}^n\mathbf{𝟏}\right)\le 1$ for every $n\in \mathbb{\mathbb{N}}$.
• $v(a)\le 1\Rightarrow v(1+a)\le 1$ for all $a\in F$.
• $v(a+b)\le \max \{v(a),v(b)\}$ 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.Template:Sfnp

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 Template:Mvar over a field Template:Mvar, represented as Template:Math, is called an absolute value, but more usually a norm, if it satisfies the following axioms:

For all Template:Mvar in Template:Mvar, and Template:Math, Template:Math in Template:Mvar,

$\Vert \mathbf{𝐯}\Vert \ge 0$	Non-negativity
$\Vert \mathbf{𝐯}\Vert =0⟺\mathbf{𝐯}=0$	Positive-definiteness
$\Vert a\mathbf{𝐯}\Vert =\left|a\right|\Vert \mathbf{𝐯}\Vert$	Absolute homogeneity or positive scalability
$\Vert \mathbf{𝐯}+\mathbf{𝐮}\Vert \le \Vert \mathbf{𝐯}\Vert +\Vert \mathbf{𝐮}\Vert$	Subadditivity or the triangle inequality

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

In the case of Euclidean space ${\mathbb{ℝ}}^n$, the function defined by

$\Vert (x_1,x_2,\dots ,x_n)\Vert =\sqrt{{\sum }_{i=1}^n{x}_i^2}$

is a norm called the Euclidean norm. When the real numbers $\mathbb{ℝ}$ are considered as the one-dimensional vector space ${\mathbb{ℝ}}^1$, the absolute value is a norm, and is the Template:Mvar-norm (see L^p space) for any Template:Mvar. In fact the absolute value is the "only" norm on ${\mathbb{ℝ}}^1$, in the sense that, for every norm Template:Math on ${\mathbb{ℝ}}^1$, Template:Math.

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 ${\mathbb{ℝ}}^2$.

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 $\mathbb{ℝ}$, complex numbers $\mathbb{ℂ}$, and quaternions $\mathbb{ℍ}$ 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).

See also

• Least absolute values

References

• Template:Cite report
• Nahin, Paul J.; An Imaginary Tale; Princeton University Press; (hardcover, 1998). Template:ISBN.

• O'Connor, J.J. and Robertson, E.F.; "Jean Robert Argand".

External links

• Template:Springer
• absolute value at PlanetMath.