Complex Numbers Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Complex Numbers.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Numbers of the form $a + bi$ where $a, b$ are real and $i = \sqrt{-1}$ ; they extend the real numbers to solve $x^2 = -1$ .

Extending numbers into a second dimension to solve equations like $x^2 = -1$ .

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A complex number $a+bi$ adds a sideways imaginary axis so that $x^2=-1$ finally has a solution.

Common stuck point: The procedure for complex numbers is the easy part; the trap is forgetting $i^2 = -1$ and leaving it unsimplified. Asking "Does the problem require the square root of a negative number, $i=\sqrt{-1}$ ?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the problem require the square root of a negative number, $i=\sqrt{-1}$ ?

Worked Examples

Example 1

easy

Simplify

i^2

i^3

, and

i^4

Answer

i^2 = -1, \quad i^3 = -i, \quad i^4 = 1

First step

i^2 = -1

by definition of the imaginary unit.

Full solution

2
$i^3 = i^2 \cdot i = (-1) \cdot i = -i$ .
3
$i^4 = i^3 \cdot i = (-i) \cdot i = -i^2 = -(-1) = 1$ .

The powers of

i

cycle with period 4:

i, -1, -i, 1, i, -1, -i, 1, \ldots

Knowing this cycle allows rapid simplification of any power of

i

by finding the remainder when the exponent is divided by 4.

Example 2

medium

Multiply

(3 + 2i)(1 - i)

and write the result in standard form

a + bi

Example 3

medium

Multiply

(4 + 3i)(2 - i)

Example 4

medium

Find the product of

z = 5 + 12i

with its conjugate

\bar{z}

Example 5

hard

Find the reciprocal of

3 - 4i

a + bi

form.

Example 6

challenge

Show that

|z_1 z_2| = |z_1|\,|z_2|

for

z_1 = 1 + 2i

and

z_2 = 3 - i

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Add

(4 + 3i) + (2 - 5i)

Example 2

medium

Simplify

i^{23}

Example 3

easy

What is

i^2

Example 4

easy

Write

\sqrt{-9}

in the form

bi

Example 5

easy

Identify the real and imaginary parts of

4 + 7i

Example 6

easy

Compute

i^4

Example 7

easy

Add

(2 + 3i) + (1 + 5i)

Example 8

easy

Compute the magnitude

|3 + 4i|

Example 9

easy

Compute

i^3

Example 10

easy

Subtract

(5 + 2i) - (3 + 6i)

Example 11

medium

Multiply

(2 + i)(3 + 2i)

Example 12

medium

Compute the modulus

|5 - 12i|

Example 13

medium

Compute

i^{10}

Example 14

medium

Find the complex conjugate of

7 - 3i

and its product with the original.

Example 15

medium

Solve

x^2 = -25

over the complex numbers.

Example 16

medium

Compute

(1 + i)^2

Example 17

medium

Add

(0 + 4i) + (6 + 0i)

and give the real and imaginary parts.

Example 18

medium

|{-3} - 4i|

equal to

|3 + 4i|

? Justify.

Example 19

challenge

Divide

\frac{4 + 2i}{1 + i}

and write in

a+bi

form.

Example 20

challenge

Compute

i^{1} + i^2 + i^3 + i^4

Example 21

challenge

|a + bi| = 10

and

a = 6

with

b>0

, find

b

Example 22

medium

Multiply

i\cdot(2 + 3i)

Example 23

easy

Write

\sqrt{-16}

in the form

bi

Example 24

easy

Add

(7 + 2i) + (-3 + 4i)

Example 25

easy

Subtract

(8 - i) - (3 + 4i)

Example 26

easy

Compute the magnitude

|6 + 8i|

Example 27

easy

Find the complex conjugate of

-2 + 5i

Example 28

medium

Compute

(3 - 2i)^2

Example 29

medium

Compute

|7 - 24i|

Example 30

medium

Solve

x^2 + 16 = 0

over the complex numbers.

Example 31

medium

Solve

x^2 - 2x + 5 = 0

over the complex numbers.

Example 32

medium

Compute

(2 + i) + (3 - 4i) - (1 + 2i)

Example 33

medium

Multiply

i\cdot(4 - 5i)

Example 34

hard

Divide

\frac{3 + 2i}{1 - i}

and write in

a + bi

form.

Example 35

hard

Compute

(1 + i)^4

Example 36

hard

z = 2 + 3i

, compute

z^2 - 4z + 13

Example 37

hard

|a + bi| = 13

and

a = 5

with

b > 0

, find

b

Example 38

challenge

Find all complex

z

with

z^2 = -1 + i\sqrt{3} \cdot 0

... actually, find all complex

z

with

z^2 = 2i

← Back to Complex Numbers Formula Explained Practice Mode

Related Concepts

Real Numbers Quadratic Formula

Background Knowledge

These ideas may be useful before you work through the harder examples.

real numbersquadratic formula