Learning Objectives
Introduction
Tutorial
where
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.
In this form, a is the real number part and b is the imaginary number part.
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
.
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
a + bi and a - bi are conjugates of each other.
When you multiply complex conjugates together you get:
Whenever you have an ,
use the definition and replace it with -1.
Example
5: Divide
.
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
.
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
Principal Square RootFor 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)
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: 2i- i = i
*Complex num. in stand. form (note
real num.
part is 0)
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
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.
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