# Absolute Value of a Complex Number

How to calculate a complex number's absolute value

The absolute value of a real number is its distance from 0 on the number line. A complex number's absolute value 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 from zero on the complex number plane.

In general

### Illustrated Example

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

### Practice Problems

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

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

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

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

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