Complex Numbers

Notes, Lesson 3.2
Complex Numbers

In our algebra travels so far both in Algebra I and thus far in Algebra II the sets of all the numbers which we have been studying, manipulating, and calculating with could be illustrated with the diagram below. Our "universal set" was the set of Real Numbers. With our work in the imaginary number area, we have changed this picture to look like the second illustration:

Remember in the above diagram we studied the following seperate sets which combined to become the set of Real Numbers. The following table lists each set, its definition and some examples from that set.

ABBR.	NAME	DEFINITION	EXAMPLES
N	Naturals
W	Whole Numbers
IN	Integers
Q	Rational Numbers
IR	Irrational Numbers
R	Real Numbers	IR Q	Any of the above

With imaginary numbers, we now have this alterned image of the set of numbers which we study in algebra. The new "universal set" is the set of Complex Numbers (C). This new universal set is made by doing a Union of the Real numbers with the Imaginary numbers (IM). You should also notice that the Pure Imaginary Numbers (PI) are shown as a subset of the Imaginary Numbers. Pure Imaginary numbers include i, but have not real number added on such as in 19+7i.

We now need to work on addition, subtraction, and multiplication of complex numbers. Even though i is not a variable (It is specifically the square root of -1), in algebraic operations, it "acts" that way.

Addition Example:

Simplify: (3+4i) + (2-9i)	Given Problem.
3+2 +4i-9i	Using the communtative property for addition, we can rearrange the terms to our liking, putting like terms together.
5-5i	Simplify and put in a+bi form (the standard form for complex numbers).

Subtraction Example:

Simplify: (12-7i) - (2-11i)	Given Problem.
12-2 -7i+11i	Remember that subtraction of a quantity requires you to subtract every item in the quantity.
10+4i	Simplify and put in a+bi form (the standard form for complex numbers).

Muliplication Example:

Simplify: (17+8i)(6-9i)	Given Problem.
102-153i+48i-72	Use Handshaking to multiply.
102-153i+48i+72	Because =-1
174-105i	Simplify and put in a+bi form (the standard form for complex numbers).