# Complex Numbers

In algebra, there are two

types of numbers: real numbers and imaginary numbers. Real numbers refer to any

ordinary number (e.g. 1, 2, 3 . . .) while imaginary numbers are . . . well . .

. imaginary! They don’t really exist, they are represented by a real number with

the letter **i** next to it. For example, **3i** is an imaginary number.

Complex numbers are those consisting of a real part and an imaginary part, i.e.

where ** a** is the real part and

**is the imaginary part.**

*b*i
Imaginary numbers are called so because they lie in the imaginary plane, they arise

from taking square

roots of negative numbers. The **i** on an imaginary number is equal

to the square root of negative one, i.e.

The **i** was introduced in order to simplify the problem of taking square roots

of negative numbers. For example, if we can find the square root of negative nine

as follows:

using properties of square roots, the above becomes

and since

is represented by **i**

## Properties of Imaginary Numbers

## Addition

Imaginary numbers behave like ordinary numbers when it comes to addition and subtraction:

## Multiplication

From the section on square roots, you should know that the following is true:

Therefore, it should follow that the following should also be true:

since **i** = -1, and

## Exponents

For any even number n, the following is always true

if an only if the following is also true

For example, given n = 4, an even number:

Conversely, if

is an odd number, then the following is true:

For example; given

where n = 6

then

For any odd number **m** greater than 1, the following is always true:

Whether **i** is positive or negative depends on the value of m. When working

with with odd number powers of **i**, you always split the powers into a sum

of even and odd numbers. For example:

which is the same as

The even part of the exponent determines whether **i** is positive or negative

as determined in the previous property.

## Division

Imaginary numbers can be divided just as any other number if there is only one term:

If there are two terms divided by two terms, we use the **complex conjugate**

To evaluate the following complex number, we multiply by the complex conjugate over itself.

We multiply by the complex conjugate of the denominator to eliminate the complex number

and make it a real constant.

As mentioned earlier, complex numbers consist of both a real and an imaginary part.

Any imaginary number can also be considered as a complex number with the real part

as zero, i.e.

It is important to remember that the real and imaginary parts of the complex number

do no interact directly, for example:

When adding or subtracting complex numbers, add the real part to the real part and

the imaginary part to the imaginary part:

Multiplication and division can be done on a complex number using either a real

or imaginary number, i.e.

It is important to remember that when writing a complex or imaginary number, do

not write the imaginary part in the denominator like this:

In such situations, we rationalize the denominator to become:

For more on rationalization, refer to the section on rationalization.

## Complex Numbers Examples

## Example 1

Solve the following

**Step 1**

**Step 2**

**Step 3**

remember that **i** x **i** = -1

**Step 4**

## Example 2

Evaluate the following:

**Step 1**

This example serves to emphasize the importance of exponents on **i**. The first

step is to inspect all the exponents and apply the properties we listed above.

**Step 2**

**Step 3**

**Step 4**

putting it all together

## Tutoring

Looking for someone to help you with algebra? At Wyzant, connect with algebra tutors and math tutors nearby. Prefer to meet online? Find online algebra tutors or online math tutors in a couple of clicks.