# How do you find abs( 9+i )?

Mar 23, 2016

$\left\mid 9 + i \right\mid = \sqrt{82} \approx 9.055385$

#### Explanation:

$\left\mid a + b i \right\mid$ is essentially the distance between $a + b i$ and $0$ in the Complex plane.

From Pythagoras, we get the distance formula and hence find:

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

Another way of expressing this is that $\left\mid z \right\mid = \sqrt{z \overline{z}}$ (where $\overline{z}$ means the Complex conjugate of $z$).

To see this, notice that:

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

In our example,

$\left\mid 9 + i \right\mid = \sqrt{{9}^{2} + {1}^{2}} = \sqrt{81 + 1} = \sqrt{82} \approx 9.055385$