# 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