Complex Numbers

Types of Numbers

There are two major types of numbers: Real and Imaginary

A real number is any number that can be found on the number line. 

This includes both rational and irrational numbers.

A rational number is defined as any number that can be represented as a fraction,

or a ratio as its root word would suggest. A number is rational if  it has definite ending point when represented as a decimal, or it repeats a pattern.

There are many different classifications of numbers that are considered rational. Integers, which are the numbers on a number line, whole numbers, which are all positive integers and zero, and then natural numbers, which are all positive integers not including zero, are all of the different kinds of rational numbers.

Irrational numbers are numbers that do not have a definite ending point nor a pattern.

Then, there are imaginary numbers.

Imaginary numbers contain the imaginary unit i.

Complex numbers are imaginary numbers added to real numbers.

Complex numbers are represented in standard form, as shown below

For example,

Should be shown as

Applications of Imaginary Numbers

Imaginary numbers vary from real numbers in that they do not behave the same way in many different situations. For example, if one were to attempt to graph 

They would use the zero product property to find that

Now, they cannot graph this function on a standard Cartesian plane. They must use the complex coordinate plane because that x value does not exist in the number line.

The x-axis of the complex coordinate plane is measured by real numbers:the a values. 

The y-axis is measured by the bi value.

In the example, the point 2+3i is found by going to 3i in the y-axis, and 2 in the x-axis.  

Rationalizing Imaginary Denominators

Like square roots in the denominator must be rationalized, meaning shifted to the numerator, imaginary number must also be rationalized.

Rationalizing is done by multiplying the ratio by its denominator's conjugate over itself.

A complex conjugate is a number that eliminates the middle terms of the product of two complex numbers

The conjugate of 

is

because

is

Which removes the i from the denominator

Finding Value of i to Any Power

One can also find i to any power because

For example,

Because

Citations

Larson, Ron, Robert P. Hostetler, Bruce H. Edwards, and David E. Heyd.Precalculus with Limits: A Graphing Approach. Boston: Houghton Mifflin, 1997. Print.

"Complex Numbers: Introduction." Complex Numbers: Introduction. N.p., n.d. Web. 13 Jan. 2015.

"Types of Numbers." Provo College Library. N.p., 23 July 2013. Web. 16 Jan. 2015.