Algebra II

Imaginary and Complex Numbers

Lesson

The imaginary unit $i$ is defined by:

i = \sqrt{-1}, \quad i^2 = -1

With $i$ , we can take the square root of any negative number: $\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i$ .

Complex numbers

A complex number has a real part and an imaginary part: $a + bi$ .

Operations with complex numbers

Add/subtract: combine real parts and imaginary parts separately, like adding terms.
Multiply: use FOIL like normal binomials. Whenever you see $i^2$ , replace it with $-1$ .
Powers of i cycle: $i^1 = i, \; i^2 = -1, \; i^3 = -i, \; i^4 = 1$ , then it repeats.

Worked example 1 — adding

(3 + 2i) + (5 - 4i) = 8 - 2i

Worked example 2 — multiplying

(2 + 3i)(1 - i)

FOIL:

= 2 - 2i + 3i - 3i^2

Replace $i^2$ with $-1$ :

= 2 + i + 3 = 5 + i

How to type your answer

Write in the form $a + bi$ or $a - bi$ . Real part first, then imaginary. No spaces. If the answer is purely real or purely imaginary, write just that part. Examples: 5+i, 8-2i, 3i, -1, -2-5i.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

i^2

Problem 2

i^4

Problem 3

(2 + i) + (3 + 2i)

Problem 4

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

Practice

Standard problems matching the lesson.

Problem 5

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

Problem 6

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

Problem 7

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

Problem 8

(1 - 4i) - (3 - 7i)

Problem 9

i^3

Problem 10

i^{10}

Problem 11

(2 + 3i)(1 - i)

Problem 12

(1 + 2i)(3 + i)

Problem 13

(4 - i)(2 + i)

Problem 14

(3 + i)(3 - i)

Problem 15

(1 + i)^2

Problem 16

(2 - i)^2

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

i^{17}

Problem 18

(5 + 2i)(5 - 2i)

Problem 19

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

Problem 20

\sqrt{-25}

Problem 21

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

Problem 22

(1 + i)^3