Complex Numbers

An imaginary number is an even root of a negative number. The definition of an imaginary unit is:

i² = -1 or √(-1)= i

Note: The imaginary unit has the special property that successive powers run through a cycle of values.

Any power of i that is a multiple of 4 (i.e. 4, 8, 12, 16, 20) is equal to 1.

Example 1 – reduce i²¹

i²¹
i²⁰ * i¹ =
1 * i = 1i

Example 2 – reduce i¹⁵

i¹⁵
i¹² * i³
i¹² * i²* i¹
1 * -1 * i
-1i

Here are some of the examples of imaginary numbers -2i, 1.05i, 2i/5, etc.

Complex numbers are written in the form a + bi

a = real part
b = imaginary part

3 + 5i

Real part – 3
Imaginary part – 5

5i

Real part – 0
Imaginary part – 5

3

Real part – 3
Imaginary part – 0

Example 1 – Add -3 + 5i and 7 – 11i.

Combine like terms.

-3 + 5i + ( 7 – 11i)
-3 + 5i + 7 – 11i
4 -6i

Example 2 – Find the product (3 – 5i)(5 + 4i).

FOIL the factors

(3 – 5i)(5 + 4i)
15 – 25i + 12i – 20i²
15 – 13i – 20*(-1)
15 – 13i + 20

Combine like terms.

15 – 13i + 20
35 – 13i

*Note: Negative square roots, such as √-16, 3√-4, and 5√-36 are invalid, so we must convert them to imaginary form.

Example 1 – Express √-16 in terms of i.

To remove the negative sign, factor an i from the term.
i√16
4i

Example 2 – Express 3√-4 in terms of i.

Factor the i from the term. Simplify from there.
3 * i√4
3* 2 * i
6i

Example 3 – Express 5√-36 in terms of i.

5* i * √36
5* i* 6
30i

The operations of addition, subtraction, multiplication, and division with radicals, apply to imaginary numbers.

Example 1 – Multiply √-5 * √-25

√-5 * √-25
i√5 * i√25
i 2* √5 * 5
-1 * 5 * √5
-5√5

Example 2 – Solve 5√-36 – 2√-36

5√-36 – 2√-36
5* i * √36 – 2* i * √36
5*i *6 – 2* i * 6
30i – 12i
18i

Graphing Imaginary Numbers

Ordinary points are graphed on the standard x-y coordinate plane, whereas imaginary numbers are graphed in the imaginary plane. The x-axis represents the real numbers, and the y-axis represents the imaginary numbers. Below is a graphical illustration of plotting imaginary expressions.