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.
absolute value of real number 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.

In general , the formula is:


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

Illustrated Example

Absolute Value of Complex Number

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

Problem 1

Calculate $$ |5+ 12i| $$

Practice Problem 1
Problem 2

What is $$ |6 + 8i| $$?

Loading absolute value problem
Problem 3

Calculate the value of $$ |3 + 2i| $$

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

Problem 4

Calculate the value of $$ |3 \red{-} 2i| $$

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

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.

Problem 5

Calculate the value of $$ |-5i- 3| $$

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

Problem 6

What is $$ |-x - ci| $$

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


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