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 Absolute Value Formula

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 plane.

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

The only difference between this one and problem 2 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.

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

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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