# How do you simplify abs(-14)?

Dec 22, 2015

$\left\mid - 14 \right\mid = 14$

#### Explanation:

If $x$ is a Real number then $\left\mid x \right\mid$ can be defined as follows:

$\left\mid x \right\mid = \left\{\begin{matrix}x & \mathmr{if} x \ge 0 \\ - x & \mathmr{if} x < 0\end{matrix}\right.$

In our case $- 14 < 0$ so $\left\mid - 14 \right\mid = - \left(- 14\right) = 14$

If $z$ is a Complex number then $\left\mid z \right\mid$ can be defined as follows:

$\left\mid z \right\mid = \sqrt{z \overline{z}}$

where $\overline{z}$ is the Complex conjugate of $z$.

For example:

$\left\mid 3 + 4 i \right\mid = \sqrt{\left(3 + 4 i\right) \left(3 - 4 i\right)} = \sqrt{{3}^{2} + {4}^{2}} = \sqrt{9 + 16}$

$= \sqrt{25} = 5$

This gives the same value as the above definition for Real numbers, if $z$ happens to be Real.