**Learning Objectives**

After completing this tutorial, you should be able to:

- Take the principle square root of a negative number.
- Write a complex number in standard form.
- Add and subtract complex numbers.
- Multiply complex numbers.
- Divide complex numbers.

** Introduction**

In this tutorial we will be looking at imaginary and
complex numbers.
Imaginary numbers allow us to take the square root of negative
numbers.
I will take you through adding, subtracting, multiplying and dividing
complex
numbers as well as finding the principle square root of negative
numbers.
I do believe that you are ready to get acquainted with imaginary and
complex
numbers.

** Tutorial**

where

This is the definition of an imaginary number.

From here on out, anytime that you have the square
root of -1 you
can simplify it as *i* and anytime you
have you can simplify it as -1.

Complex numbers are made up of a real number part and
an imaginary
number part.

In this form, ** a is the
real number part** and

Note that either one of these parts can be 0.

**An example of a complex number written in standard
form is**

.

**if and only if a = c AND b = d.**

** Example
1**: Add .

** Example
2**: Subtract .

If you need a review on multiplying polynomials, go to **Tutorial
6: Polynomials.**

Add real numbers together and imaginary numbers
together.

Whenever you have an , use the definition and replace it with -1.

** Example
3**: Multiply .

**AND**

**Step 2: Simplify
the expression.**

*** i squared
= -1**

** Example
4**: Multiply .

**AND**

**Step 2: Simplify
the expression.**

***Combine imaginary numbers**

*** i squared
= -1**

You find the conjugate of a binomial by changing the
sign that is between
the two terms, but keep the same order of the terms.

*a *+* bi* and *a* - *bi *are conjugates of each other.

Keep in mind that as long as you multiply the numerator
and denominator
by the exact same thing, the fractions will be equivalent.

**When you multiply complex conjugates together you
get: **

Add real numbers together and imaginary numbers
together.

Whenever you have an , use the definition and replace it with -1.

** Example
5**: Divide .

In general the conjugate of *a* + *bi* is *a* - *bi *and
vice
versa.

So what would the conjugate of our denominator be?

**It looks like the conjugate is **.

**AND**

**Step 3: Simplify the expression.**

*****

*** i squared
= -1**

***Divide each term of num. by 5**

***Complex num. in stand. form**

** Example
6**: Divide .

In general the conjugate of *a* + *bi* is *a* - *bi *and
vice
versa.

So what would the conjugate of our denominator be?

**It looks like the conjugate is **.

**AND**

**Step 3: Simplify the expression.**

*****

*** i squared
= -1**

***Complex num. in stand. form**

**For any positive real number b,
the principal
square root of the negative number, -b, is defined by**

** Example
7**: Simplify .

***Complex num. in stand. form
(note real num.
part is 0)**

In other words use the definition of principal square
roots of negative
numbers before performing any operations.

** Example
8**: Perform the indicated operation. Write answer in
standard
form.

**AND**

**Step 3: Write
the final answer in standard form.**

***The square root of 4 is 2**

***Subtract like radicals: 2 i- i = i**

** Example
9**: Perform the indicated operation. Write answer in
standard
form.

**AND**

**Step 3: Write
the final answer in standard form.**

*** i squared
= -1**

***Complex num. in stand. form**

** Example
10**: Perform the indicated operation. Write answer in
standard
form.

**AND**

**Step 3: Write
the final answer in standard form.**

***The square root of 25 is 5**

***Divide each term of num. by 5**

***Complex num. in stand. form**

** Example
11**: Perform the indicated operation. Write answer in
standard
form.

**AND**

**Step 3: Write
the final answer in standard form.**

** Practice Problems**

These are practice problems to help bring you to the
next level.
It will allow you to check and see if you have an understanding of
these
types of problems. **Math works just like
anything
else, if you want to get good at it, then you need to practice
it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.**
In fact there is no such thing as too much practice.

To get the most out of these, **you should work the
problem out on
your own and then check your answer by clicking on the link for the
answer/discussion
for that problem**. At the link you will find the answer
as well as any steps that went into finding that answer.

Practice Problems 1a - 1i:Perform the indicated operation.Write the answer in standard form.

** Need Extra Help on these Topics?**

**Go to Get
Help Outside the
Classroom found in Tutorial 1: How to Succeed in a Math Class for
some
more suggestions.**

Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.

Last revised on Dec. 15, 2009 by Kim Seward.

All contents copyright (C) 2002 - 2010, WTAMU and Kim Seward.
All rights reserved.