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
Imaginary numbers behave like ordinary numbers when it comes to addition and subtraction:
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
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
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.
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
Solve the following
remember that i x i = -1
Evaluate the following:
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.
putting it all together