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 bi is the imaginary part.

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

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