### Illustrated **Example**

To find the absolute value of the complex number, 3 + 4*i*,
we find the distance from
zero to
that number on the
complex plane.

As you might be able to tell, the final solution is basically the distance formula !

Debug

The absolute value of a real number like $$|4| $$ is its distance from 0 on the number line.

The absolute value of complex number is also a measure of its distance from zero. However, instead of measuring this distance on the number line, a complex number's absolute value is measured on the complex number plane.

$
\text{ The General Formula}
$

$|a + bi| = \sqrt{a^2 + b^2 } $

To find the absolute value of the complex number, 3 + 4*i*,
we find the distance from
zero to
that number on the
complex plane.

As you might be able to tell, the final solution is basically the distance formula !

$$ |3 + 2i| = \sqrt{3^2 + (2)^2} \\ = \sqrt{9 + 4} \\ = \sqrt{13} $$

$$ |3 - 2i| = \sqrt{3^2 + (\red{-}2)^2} \\ = \sqrt{9 + \red{4}} \\ = \sqrt{13} $$

The only difference between this question and prior problem is that the $$ 2i $$ has now become $$ -2i $$. However, since we are squaring that term, the negative sign has no effect and you end up with the exact same answer.

$$ |-5 -3i| = \sqrt{(-3)^2 + (-5)^2} \\ = \sqrt{9 + 25} \\ = \sqrt{34} $$

$$ | -x - ci| = \sqrt{(-x)^2 + (-c)^2} \\ = \sqrt{x^2 + c^2} $$

Further Reading