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

i^{2} = -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 i ^{21}**

i^{21}

i^{20} * i^{1} =

1 * i = 1i

**Example 2 – reduce i ^{15}**

i^{15}

i^{12} * i^{3}

i^{12} * i^{2}* i^{1}

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 – 20i^{2}

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.