Download 7.7

Chapter 7
Roots,
Radicals, and
Complex
Numbers
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-1
1
Chapter Sections
7.1 – Roots and Radicals
7.2 – Rational Exponents
7.3 – Simplifying Radicals
7.4 – Adding, Subtracting, and Multiplying
Radicals
7.5 – Dividing Radicals
7.6 – Solving Radical Equations
7.7 – Complex Numbers
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-2
2
§ 7.7
Complex Numbers
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-3
3
Recognize a Complex Number
An imaginary number is a number such as 5.
It is called imaginary because when imaginary
numbers were first introduced, many
mathematicians refused to believe they existed!
Every imaginary number has 1 as a factor.
The 1 , called the imaginary unit, is
denoted by the letter i.
i  1
For any positive number n,
n i n
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-4
4
Recognize a Complex Number
Imaginary Unit
i  1
Square Root of a Negative Number
For any positive real number n,
 n   1 n   1 n  i n
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-5
5
Recognize a Complex Number
Complex Number
Every number of the form
a  bi
Where a and b are real numbers and I is the imaginary
unit, is a complex number.
Every real number and every imaginary number
are also complex numbers. A complex number has
two parts: a real part, a, and an imaginary part, b.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-6
6
Recognize a Complex Number
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-7
7
Recognize a Complex Number
Complex numbers can be added, subtracted,
multiplied, and divided. To perform these operations,
we use the definitions that i   1 and i 2  1.
Definition of i2
If i   1 , then
i  1
2
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-8
8
Add and Subtract Complex Numbers
To Add or Subtract Complex Numbers
1. Change all imaginary numbers to bi form.
2. Add (or subtract) the real parts of the complex
numbers.
3. Add (or subtract) the imaginary parts of the
complex numbers.
4. Write the answer in the form a + bi.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-9
9
Add and Subtract Complex Numbers
Example Add (9  15i )  (6  2i)  18
(9  15i )  (6  2i )  18  9  15i  6  2i  18
 9  6  18  15i  2i
 21  13i
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-10
10
Multiply Complex Numbers
To Multiply Complex Numbers
1. Change all imaginary numbers to bi form.
2. Multiply the complex numbers as you would
multiply polynomials.
3. Substitute –1 for each i2.
4. Combine the real parts and the imaginary parts.
Write the answer in a + bi form.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-11
11
Multiply Complex Numbers
Example
Multiply 5i (6  2i )
5i (6  2i )  5i (6)  5i (2i )
 30i  10i 2
 30i  10(1)
 30i  10 or
10  30i
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-12
12
CAUTION!
4 2 ?
 4   2  i 4 i 2 
2i
2
2  2 2
4 2  8
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-13
13
Divide Complex Numbers
The conjugate of a complex number a + bi is a – bi.
For example,
Complex Number
Conjugate
3  7i
3  7i
1 i 3
2i (or 0  2i )
1 i 3
 2i (or 0  2i )
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-14
14
Divide Complex Numbers
To Divide Complex Numbers
1. Change all imaginary numbers to bi form.
2. Multiply both the numerator and denominator by
the conjugate of the denominator.
3. Write the answer is a + bi form.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-15
15
Divide Complex Numbers
Example
Divide
9i
i
9  i  i (9  i )(i )
 
i i
 i2
 9i  i 2

2
i
 9i  (1)

 (1)
 9i  1

1
 1  9i
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-16
16
Find Powers of i
The successive powers of i rotate through the four
values of i, -1, -i, and 1.
in = i if
n = 1, 5, 9, …
in = 1 if
n
= 4, 8, 12, …
in = -1 if
n = 2, 6, 10, …
in = -i if
n = 3, 7, 11, …
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-17
17