Absolute value

In mathematics, the absolute value of a real number ${\displaystyle x}$, written as ${\displaystyle |x|}$or ${\displaystyle {\text{abs}}(x)}$, is the value of ${\displaystyle x}$when the sign is dropped (or ${\displaystyle x}$without its sign). That is, ${\displaystyle |x|=x}$for a positive ${\displaystyle x}$, ${\displaystyle |x|=-x}$for a negative ${\displaystyle x}$, and ${\displaystyle |0|=0}$

For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3 (${\displaystyle |x|=|-x|}$). The absolute value of a real number may be thought of as its distance from zero. It can be defined as follows:

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

Because a distance is always positive, the absolute value of a number is always positive.

Similarly, the absolute value of a complex number may be thought of as its distance from the origin.

It is defined by the equation

${\displaystyle |a+bi|={\sqrt {a^{2}+b^{2}}}}$

For any real number ${\displaystyle x}$, its absolute value is denoted by ${\displaystyle |x|}$(a vertical bar on each side of the quantity), and is defined as

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

The absolute value of ${\displaystyle x}$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 continuous everywhere.

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:

${\displaystyle |a|={\sqrt {a^{2}}}}$

which is another definition of absolute value.

The absolute value has the following four main properties:

Other important properties of the absolute value include:

Two other useful properties related to inequalities are:

${\displaystyle |a|\leq b\Leftrightarrow -b\leq a\leq b}$

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

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

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

For a complex number ${\displaystyle z=a+bi}$, where ${\displaystyle a}$is the real part of ${\displaystyle z}$and ${\displaystyle b}$is the imaginary part of ${\displaystyle z}$,

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

• Magnitude (mathematics)