# Absolute value

nonnegative number with the same magnitude as a given real number

In mathematics, the absolute value or modulus of a real number x, written as |x| or ${\displaystyle {\text{abs}}(x)}$, is the non-negative value of x when the sign is dropped.[1][2][3] That is, |x| = x for a positive x, |x| = −x for a negative x (in which case x is positive), and |0| = 0.

The graph of the absolute value function for real numbers.

For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a real number may be thought of as its distance from zero. It can be defined as follows:

${\displaystyle {\begin{cases}\ \;\,\ \ x&\mathrm {if} \ x\geq 0\\\ \;\,-x&\mathrm {otherwise} \end{cases}}}$

Similarly, the absolute value (or modulus) of a complex number may be thought of as its distance from the origin. It is defined by the equation[3]

${\displaystyle |x+iy|={\sqrt {x^{2}+y^{2}}}\ }$

## Properties

### Real numbers

For any real number x, the absolute value of x is denoted by | x | (a vertical bar on each side of the quantity),[1] and is defined as[4]

${\displaystyle |x|={\begin{cases}x,&{\mbox{if }}x\geq 0\\-x,&{\mbox{if }}x<0.\end{cases}}}$

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. The absolute value of the difference of two real numbers is the distance between them.

In calculus, the absolute value function is differentiable except at 0. It is everywhere continuous.

In linear algebra, the norm of a vector is defined similarly as the distance from the tip of the vector to the origin. This is similar to the way the absolute value of a complex number is defined.

The square-root notation without sign represents the positive square root. So, it follows that

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

which is sometimes used as a definition of absolute value.[5]

The absolute value has the following four main properties:

 ${\displaystyle |a|\geq 0}$ (2) Non-negativity ${\displaystyle |a|=0\iff a=0}$ (3) Positive-definiteness ${\displaystyle |ab|=|a||b|\,}$ (4) Multiplicativeness ${\displaystyle |a+b|\leq |a|+|b|}$ (5) Subadditivity

Other important properties of the absolute value include:

 ${\displaystyle ||a||=|a|\,}$ (6) Idempotence (the absolute value of the absolute value is the absolute value) ${\displaystyle |-a|=|a|\,}$ (7) Symmetry ${\displaystyle |a-b|=0\iff a=b}$ (8) Identity of indiscernibles (equivalent to positive-definiteness) ${\displaystyle |a-b|\leq |a-c|+|c-b|}$ (9) Triangle inequality (equivalent to subadditivity) ${\displaystyle |a/b|=|a|/|b|{\mbox{ (if }}b\neq 0)\,}$ (10) Preservation of division (equivalent to multiplicativeness) ${\displaystyle |a-b|\geq ||a|-|b||}$ (11) (equivalent to subadditivity)

Two other useful properties related to inequalities are:

${\displaystyle |a|\leq b\iff -b\leq a\leq b}$
${\displaystyle |a|\geq b\iff a\leq -b{\mbox{ or }}b\leq a}$

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

 ${\displaystyle |x-3|\leq 9}$ ${\displaystyle \iff -9\leq x-3\leq 9}$ ${\displaystyle \iff -6\leq x\leq 12}$

Diagram showing ${\displaystyle |z|}$ , the modulus of z.

### Complex numbers

For a complex number ${\displaystyle z=x+iy}$ , where x is the real part of z and y is the imaginary part of z,[3]

${\displaystyle |z|=|x+iy|={\sqrt {x^{2}+y^{2}}}\ }$

## References

1. "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-28.
2. "Absolute Value". www.mathsisfun.com. Retrieved 2020-08-28.
3. Weisstein, Eric W. "Absolute Value". mathworld.wolfram.com. Retrieved 2020-08-28.
4. Mendelson, p. 2.
5. Stewart, James B. (2001). Calculus: concepts and contexts. Australia: Brooks/Cole. ISBN 0-534-37718-1. Cite has empty unknown parameter: |coauthors= (help), p. A5