Magnitude of Complex Numbers - Examples and Practice Problems

The magnitude of a complex number is equal to its distance from the origin in the complex plane. The process of finding the magnitude of a complex number is very similar to the process of finding the distance between two points.

Here, we will learn how to calculate the magnitude of complex numbers using a formula. In addition, we will look at several examples with answers to master the application of the formula completely.

ALGEBRA

Relevant for…

Learning about the magnitude of complex numbers with examples.

See examples

Contents

ALGEBRA

Relevant for…

Learning about the magnitude of complex numbers with examples.

See examples

How to calculate the magnitud of complex numbers?

The magnitude of a complex number can be calculated using a process similar to finding the distance between two points. Recall that the distance between two points can be found using the formula:

$d= \sqrt{{{(x_{2}-x_{1})}^2}+{{(y_{2}-y_{1})}^2}}$

If we want to find the distance from the origin in the Cartesian plane, this formula simplifies to:

$d= \sqrt{{{x}^2}+{{y}^2}}$

In the complex plane, the x-axis represents the real axis and the y-axis represents the imaginary axis. If we have a complex number in the form $z=a+bi$ , the formula for the magnitude of this complex number is:

In this formula, a is our real component and b is our imaginary component. Furthermore, we denote the magnitude of a complex number as $|z|$ .

Magnitude of complex numbers – Examples with answers

The process used to calculate the magnitude of complex numbers mentioned above is used to solve the following examples. Each example has its respective solution, but it is recommended that you try to solve the exercises yourself before looking at the answer.

EXAMPLE 1

What is the magnitude of the number $z=3+4i$ ?

Solution

We simply use the formula for the magnitude of complex numbers and plug in the values of $a=3$ and $b=4$ :

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

$=\sqrt{{{3}^2}+{{4}^2}}$

$=\sqrt{9+16}$

$=\sqrt{25}$

$|z|=5$

EXAMPLE 2

What is the magnitude of the number $z=-5+6i$ ?

Solution

We have to use the formula for the magnitude of complex numbers with the values of $a=-5$ and $b=6$ :

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

$=\sqrt{{{(-5)}^2}+{{6}^2}}$

$=\sqrt{25+36}$

$|z|=\sqrt{61}$

EXAMPLE 3

Find the magnitude of $z=-4-7i$ .

Solution

Here, we have the values $a=-4$ and $b=-7$ :

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

$=\sqrt{{{(-4)}^2}+{{(-7)}^2}}$

$=\sqrt{16+49}$

$|z|=\sqrt{65}$

Start now: Explore our additional Mathematics resources

EXAMPLE 4

Find the magnitude of $z=6+2i$ .

Solution

We can use the formula for the magnitude of complex numbers and with the values $a=6$ and $b=2$ :

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

$=\sqrt{{{6}^2}+{{2}^2}}$

$=\sqrt{36+4}$

$=\sqrt{40}$

$=\sqrt{4\times 10}$

$|z|=2\sqrt{10}$

EXAMPLE 5

What is the magnitude of the number $z=10-5i$ ?

Solution

We simply use the formula for the magnitude of complex numbers and plug in the values of $a=10$ and $b=-5$ :

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

$=\sqrt{{{10}^2}+{{(-5)}^2}}$

$=\sqrt{100+25}$

$=\sqrt{125}$

$=\sqrt{25\times 5}$

$|z|=5\sqrt{5}$

Magnitude of complex numbers – Practice problems

Solve the following problems to practice what you have learned about the magnitude of complex numbers. If you need help with this, you can look at the solved examples above.