# How do you find abs( -4+2i )?

May 26, 2016

$| - 4 + 2 i | = 2 \sqrt{5} \cong 4.5$

#### Explanation:

We have the complex number

$c = - 4 + 2 i$

There are two equivalent expressions for the magnitude of an imaginary number, one in terms of the real and imaginary parts and

$| c | = + \sqrt{\mathbb{R} e {\left(c\right)}^{2} + I m {\left(c\right)}^{2}}$ ,

and another in terms of the complex conjugate

        $= + \sqrt{c \cdot \overline{c}}$ .

I'm going to use the first expression because it's simpler, in certian cases the 2nd may be more useful.

We need the real part and imaginary parts of $- 4 + 2 i$
$\mathbb{R} e \left(- 4 + 2 i\right) = - 4$
$I m \left(- 4 + 2 i\right) = 2$
$| - 4 + 2 i | = \sqrt{{\left(- 4\right)}^{2} + {\left(2\right)}^{2}} = \sqrt{16 + 4} = \sqrt{20} = 2 \sqrt{5} \cong 4.5$