Transcript Ch 1.3 Complex Numbers ppt
10 TH EDITION
COLLEGE ALGEBRA
LIAL HORNSBY SCHNEIDER
1.3 - 1
1.3
Complex Numbers
Basic Concepts of Complex Numbers Operations on Complex Numbers
Basic Concepts of Complex Numbers
There are no real numbers for the solution of the equation
x
2 1.
To extend the real number system to include such numbers as, 1, the number
i
is defined to have the following property;
i
2 1.
Basic Concepts of Complex Numbers
So…
i
1 The number
i
is called the
imaginary unit
.
Numbers of the form
a
+
bi
, where
a
and
b
are real numbers are called
complex numbers
.
In this complex number,
a
is the
real part
and
b
is the
imaginary part
.
Complex numbers
a
+
bi
,
a
and
b
real Nonreal complex numbers
a
+
bi
,
b
≠ 0 Real numbers
a
+
bi
,
b
= 0 Irrational numbers Rational numbers Integers Non integers
Basic Concepts of Complex Numbers
Two complex numbers are equal provided that their real parts are equal and their imaginary parts are equal;
a
bi di
if and only if
a
c
and
b
d
Basic Concepts of Complex Numbers
For complex number
a
+
bi
, if
b
= 0, then
a
+
bi
=
a
So, the set of real numbers is a subset of complex numbers.
Basic Concepts of Complex Numbers
If
a
= 0 and
b
≠ 0, the complex number is
pure imaginary
.
A pure imaginary number or a number, like 7 + 2
i
with
a
≠ 0 and
b
≠ 0, is a
nonreal complex number
. The form
a
form
.
+
bi
(or
a
+
ib
) is called
standard
THE EXPRESSION
a
If
a
0, then
i a
.
Example 1
i a
Write as the product of a real number and
i
, using the definition of
a
.
a.
16
Solution
: 16
i
16 4
i
Example 1
i a
Write as the product of a real number and
i
, using the definition of
a
.
b.
70
Solution
: 70
i
70
Example 1
i a
Write as the product of a real number and
i
, using the definition of
a
.
c.
48
Solution
: 48
i
48
i
16 3 4
i
3 Product rule for radicals
Operations on Complex Numbers
Products or quotients with negative radicands are simplified by first rewriting
a
as
i a
for a positive number.
Then the properties of real numbers are applied, together with the fact that
i
2 1.
Operations on Complex Numbers
Caution
When working with negative radicands, use the definition…
i a
before using any of the other rules for radicands.
Operations on Complex Numbers
Caution
In particular, the rule
c d
cd
is valid only when
c
and
d
are not both negative.
36 6, while 4 6
i
2 6 so 4 9
a.
Example 2
FINDING PRODUCTS AND QUOTIENTS INVOLVING NEGATIVE RADICALS
Multiply or divide, as indicated. Simplify each answer.
7 7
Solution
: 7
First write all square roots in terms of i.
i i
2 1 7 7
i
7 2 7
i
2 = −1
b.
Example 2
FINDING PRODUCTS AND QUOTIENTS INVOLVING NEGATIVE RADICALS
Multiply or divide, as indicated. Simplify each answer.
6 10
Solution
: 6 10
i
6
i
10 1 2 15
i
2 60 1 4 15 2 15
Example 2
FINDING PRODUCTS AND QUOTIENTS INVOLVING NEGATIVE RADICALS
Multiply or divide, as indicated. Simplify each answer.
c.
20 2
Solution
: 20 2
i i
20 2 20 2 10 Quotient rule for radicals
Example 2
FINDING PRODUCTS AND QUOTIENTS INVOLVING NEGATIVE RADICALS
Multiply or divide, as indicated. Simplify each answer.
d.
48 24
Solution
: 48 24
i i
48 2 4
i
48 24
i
2
Example 3 Write 4
SIMPLIFYING A QUOTIENT INVOLVING A NEGATIVE RADICAND
128 in standard form
a
+
bi
.
Solution
: 4 12 8 4 64 2 4 8
i
2 64 8
i
Example 3 Write 4
SIMPLIFYING A QUOTIENT INVOLVING A NEGATIVE RADICAND
128 in standard form
a
+
bi
.
Solution
:
Be sure to factor before simplifying
4 4 8
i
2 2
i
2 64 8
i
2
i
4 2 Factor.
Lowest terms
Addition and Subtraction of Complex Numbers For complex numbers
a
+
bi
and
c
+
di
, (
a
bi
c
di
) (
a
c b
and (
a
bi
c
di
) (
a
c b
Example 4
ADDING AND SUBTRACTING COMPLEX NUMBERS
Find each sum or difference.
a.
(3
Solution
: ( 3 4
i
Add real parts.
Add imaginary parts.
2 6
i
) 3 ( 2 ) 4 6
i
Commutative, associative, distributive properties 1 2
i
Example 4
ADDING AND SUBTRACTING COMPLEX NUMBERS
Find each sum or difference.
b.
Solution
: 8
i
Example 4
ADDING AND SUBTRACTING COMPLEX NUMBERS
c.
Find each sum or difference.
Solution
:
i
i
Example 4
ADDING AND SUBTRACTING COMPLEX NUMBERS
d.
Find each sum or difference.
Solution
: 2
i
Multiplication of Complex Numbers
The product of two complex numbers is found by multiplying as if the numbers were binomials and using the fact that
i 2
= –1, as follows.
(
a
di
)
ac
adi
bic
bidi
FOIL
ac
adi
bci
bd i
2 Distributive property;
i
2 = – 1
ac
(
ad
bd
( 1 ) (
ac
bd
ad
Multiplication of Complex Numbers For complex numbers
a
+
bi
and
c
+
di
, (
a
di
) (
ac
bd ad
a.
Example 5
MULTIPLYING COMPLEX NUMBERS
Find each product.
Solution
: 2(3) 8
i
9
i
12
i
2
i
18
i
12 ( 1 ) FOIL
i
2 = −1
Example 5
MULTIPLYING COMPLEX NUMBERS
Find each product.
b.
(4
i
2
Solution
: (4
i
2 4 2 16 24
i
9
i
i
2 Square of a binomial
i
2
Remember to add twice the product of the two terms.
16 24
i i
2 = −1 24
i
Example 5
MULTIPLYING COMPLEX NUMBERS
Find each product.
c.
(6
Solution
: (6 6 2
i
2 Product of the sum and difference of two terms 36 36 25 61, or 61 0
i i
2 = −1 Standard form
Simplifying Powers of
i
Powers of
i
can be simplified using the facts
i
2 1 and
i
4
i
2 ( 1 ) 2 1
Example 6
SIMPLIFYING POWERS OF i
Simplify each power of
i
.
a.
i
15
Solution
: Since
i
2 = –1 and
i
4 = 1, write the given power as a product involving
i
2 example,
i
3
i
2
i
( 1 )
i
or
i
.
i
4 . For Alternatively, using
i
4
i
15
i
12
i
3 and
i
3
i
3
i
3 to rewrite
i
15 1 3 (
i i
gives
Example 6
SIMPLIFYING POWERS OF i
Simplify each power of
i
.
b.
i
3
Solution
:
i
3
i
4
i
(
i
4 ) 1
i
1
i
i
Powers of
i i
1
i i
2 1
i
3
i i
4 1
i
5
i i
6 1
i
7
i i
8 1
i
9
i i
10 1
i
11
i i
12 1, and so on.
Ex 5c. showed that… (6 61 The numbers differ only in the sign of their imaginary parts and are called
complex conjugates
.
The product of a complex number and its conjugate is always a real number.
This product is the sum of squares of real and imaginary parts.
Property of Complex Conjugates For real numbers
a
and
b
, (
a
bi
)
a
2
b
2 .
Example 7
DIVIDING COMPLEX NUMBERS
Write each quotient in standard form
a
+
bi
.
a.
3 5 2
i i
Solution
: 3 5 2
i i
(3 ( 5
i i
(5 ) ( 5
i i
) ) Multiply by the complex conjugate of the denominator in both the numerator and the denominator.
15 3
i
25 10
i
i
2 2
i
2 Multiply.
Example 7
DIVIDING COMPLEX NUMBERS
Write each quotient in standard form
a
+
bi
.
a.
3 5 2
i i
Solution
: 15 3
i
10
i
25
i
2 2
i
2 Multiply.
i
26
i
2 = −1
Example 7
DIVIDING COMPLEX NUMBERS
Write each quotient in standard form
a
+
bi
.
a.
3 5 2
i i
Solution
:
i
26 13 26 13
i
26
i
2 = −1
a
bi c c bi c
Example 7
DIVIDING COMPLEX NUMBERS
Write each quotient in standard form
a
+
bi
.
a.
3 5 2
i i
Solution
: 13 26 13
i
26
a
bi c c bi c
2 1
i
2 Lowest terms; standard form
Example 7
DIVIDING COMPLEX NUMBERS
Write each quotient in standard form
a
+
bi
.
b.
3
i
Solution
:
i
3 3 (
i
)
i
(
i
) 3
i
i
2 –
i
is the conjugate of
i.
Example 7
DIVIDING COMPLEX NUMBERS
Write each quotient in standard form
a
+
bi
.
b.
3
i
Solution
: 3
i
i
2 3
i
1
i
2 = −1(−1) = 1
i
Standard form