Download Multiply complex numbers.

Chapter 9
Section 4
9.4 Complex Numbers
Objectives
1
Write complex numbers as multiples of i.
2
Add and subtract complex numbers.
3
Multiply complex numbers.
4
Divide complex numbers.
5
Solve quadratic equations with complex number solutions.
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Complex Numbers
Some quadratic equations have no real number solutions. For example,
the numbers
4  4
,
2
are not real numbers because – 4 appears in the radicand. To ensure
that every quadratic equation has a solution, we need a new set of
numbers that includes the real numbers. This new set of numbers is
defined with a new number i, call the imaginary unit, such that
i  1
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and
i 2  1.
Slide 9.4-3
Objective 1
Write complex numbers as
multiples of i.
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Slide 9.4-4
Write complex numbers as multiples if i.
We can write numbers such as
4,
5, and 8 as multiples of
i, using the properties of i to define any square root of a negative
number as follows.
b
For any positive real number b,
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b  i b.
Slide 9.4-5
EXAMPLE 1 Simplifying Square Roots of Negative Numbers
Write
15
as a multiple of i.
Solution:
i 15
It is easy to mistake 2i for
2i , with the i under the radical. For this
reason, it is customary to write the factor i first when it is multiplied by a
radical. For example, we usually write i 2 rather than
2i.
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Slide 9.4-6
Write complex numbers as multiples if i. (cont’d)
Numbers that are nonzero multiples of i are pure imaginary
numbers. The complex numbers include all real numbers and all
imaginary numbers.
Complex Number
A complex number is a number of the form a + bi, where a and b
are real numbers. If a = 0 and b ≠ 0, then the number bi is a pure
imaginary number.
In the complex number a + bi, a is called the real part and b is
called the imaginary part. A complex number written in the form
a + bi (or a + ib) is in standard form. See the figure on the
following slide which shows the relationship among the various
types of numbers discussed in this course.
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Slide 9.4-7
Write complex numbers as multiples of i. (cont’d)
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Slide 9.4-8
Objective 2
Add and subtract complex numbers.
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Slide 9.4-9
Add and subtract complex numbers.
Adding and subtracting complex numbers is similar to adding and
subtracting binomials.
To add complex numbers, add their real parts and add their
imaginary parts.
To subtract complex numbers, add the additive inverse (or opposite).
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Slide 9.4-10
EXAMPLE 2 Adding and Subtracting Complex Numbers
Add or subtract.
Solution:
 1  8i   9  3i 
 8  5i
 6  i    5  4i 
 1  3i
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Slide 9.4-11
Objective 3
Multiply complex numbers.
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Slide 9.4-12
Multiply complex numbers.
We multiply complex numbers as we do polynomials. Since i2 = –1
by definition, whenever i2 appears, we replace it with –1.
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Slide 9.4-13
EXAMPLE 3 Multiplying Complex Numbers
Find each product.
Solution:
6i  4  3i 
1  5i 3  7i 
 18  24i
 3  15i  7i  35i
 3  8i  35
 38  8i
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2
Slide 9.4-14
Objective 4
Divide complex numbers.
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Slide 9.4-15
Write complex number quotients in standard form.
The quotient of two complex numbers is expressed in standard form
by changing the denominator into a real number.
The complex numbers 1 + 2i and 1 – 2i are conjugates. That is, the
conjugate of the complex number a + bi is a – bi. Multiplying the
complex number a + bi by its conjugate a – bi gives the real number
a2 + b2.
Product of Conjugates
 a  bi  a  bi   a2  b2
That is, the product of a complex number and its conjugate is the
sum of the squares of the real and imaginary part.
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Slide 9.4-16
EXAMPLE 4 Dividing Complex Numbers
Write the quotient in standard form.
3  4i
1 i
4i
i
Solution:
1 7
  i
2 2
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1  4i
Slide 9.4-17
Objective 5
Solve quadratic equations with
complex number solutions.
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Slide 9.4-18
Solve quadratic equations with complex solutions.
Quadratic equations that have no real solutions do have complex
solutions.
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Slide 9.4-19
EXAMPLE 5
Solving a Quadratic Equation with Complex Solutions (Square Root Property)
Solve (x – 2)2 = –64.
Solution:
x  2  64
x  2  i 64
x  2  i 64
x  2  8i
or x  2  i 64
or
x  2  8i
2  8i
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Slide 9.4-20
EXAMPLE 6
Solving a Quadratic Equation with Complex Solutions (Quadratic Formula)
Solve x2 – 2x = –26.
Solution:
x 2  2 x  26  0
a  1, b  2, c  26
b  b2  4ac
x
2a
x
  2  
 2   4 1 26 
2
2 1
2  4  104
x
2 1
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2  100
x
2
2  i 100
x
2
2  10i
x
2
x  1  5i
1  5i
Slide 9.4-21