# How do you find abs( -6-8i )?

Apr 10, 2016

$\left\mid - 6 - 8 i \right\mid = 10$

#### Explanation:

The modulus of a Complex number is essentially its distance from the number $0$ in the Complex plane.

By Pythagoras theorem that means that:

$\left\mid a + b i \right\mid = \sqrt{{a}^{2} + {b}^{2}}$

In our example,

$\left\mid - 6 - 8 i \right\mid = \sqrt{{\left(- 6\right)}^{2} + {\left(- 8\right)}^{2}} = \sqrt{36 + 64} = \sqrt{100} = 10$

Another formula for $\left\mid z \right\mid$ is:

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

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

Notice that:

$\left(a + b i\right) \overline{\left(a + b i\right)} = \left(a + b i\right) \left(a - b i\right) = {a}^{2} - {b}^{2} {i}^{2} = {a}^{2} + {b}^{2}$