﻿ How to find the absolute value of a complex number: Find the distance from 0 on the complex plane by...

# Absolute Value of a Complex Number

How to calculate a complex number's absolute value

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

In general , the formula is:

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

### Illustrated Example

To find the absolute value of the complex number, 3 + 4i, 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 !

### Practice Problems

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

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

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.

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

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

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