# How do you find abs( x+iy )?

Mar 21, 2016

$\left\mid x + i y \right\mid = \sqrt{{x}^{2} + {y}^{2}}$

#### Explanation:

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

Using the distance formula which comes from Pythagoras theorem, we have:

$\left\mid x + i y \right\mid = \sqrt{{x}^{2} + {y}^{2}}$

Notice that $\left(x + i y\right) \left(x - i y\right) = {x}^{2} - {i}^{2} {y}^{2} = {x}^{2} + {y}^{2}$

So another way of expressing this is:

$\left\mid x + i y \right\mid = \sqrt{\left(x + i y\right) \left(x - i y\right)} = \sqrt{\left(x + i y\right) \overline{\left(x + i y\right)}}$

So without explicitly splitting a Complex number $z$ into Real and imaginary parts, we can say:

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