Transcript Multiplying complex numbers

Radical Functions
Copyright © Cengage Learning. All rights reserved.
8
8.5
Complex Numbers
Copyright © Cengage Learning. All rights reserved.
Objectives
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

Identify complex numbers.
Perform arithmetic operations with complex
numbers.
Find complex solutions to equations.
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Definition of Imaginary and
Complex Numbers
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Definition of Imaginary and Complex Numbers
We have noted that the square root of a negative number is
not a real number. Today, something like
is considered
a nonreal number and is called an imaginary number.
Because these numbers were not believed to really exist,
they were called imaginary numbers.
Later they were proven to exist, and they have been shown
to be applicable in different fields of mathematics and
science.
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Definition of Imaginary and Complex Numbers
Although imaginary numbers were proven to exist, the
name had stuck by then, so we must remember that the
name imaginary does not mean that these numbers do not
exist.
In mathematics, the number
is the imaginary unit and
is usually represented by the letter i.
Using the letter i, we can represent imaginary numbers
without showing a negative under a square root.
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Definition of Imaginary and Complex Numbers
Imaginary numbers can be combined with the real numbers
into what are called complex numbers.
Any number that can be written in the form a + bi is a
complex number; a is considered the real part of a complex
number, and b is considered the imaginary part.
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Definition of Imaginary and Complex Numbers
All real numbers are considered complex numbers whose
imaginary part is equal to zero.
All imaginary numbers are also considered complex
numbers whose real part is equal to zero.
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Definition of Imaginary and Complex Numbers
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Definition of Imaginary and Complex Numbers
Complex numbers are used in many areas of mathematics
and many physics and engineering fields.
Electronics uses complex numbers to work with voltage
calculations.
The shapes of airplane wings are developed and studied
by using complex numbers.
Many areas of algebra, such as fractals and chaos theory,
can be studied by working in the complex number system.
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Definition of Imaginary and Complex Numbers
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Example 2 – Name the parts of complex numbers
For each complex number, name the real part and the
imaginary part.
a. 5 + 4i
b. 23 + 7i
c. 5i
d. 9
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Example 2 – Solution
a. 5 + 4i; Real part = 5; imaginary part = 4.
b. –3 + 7i; Real part = –3; imaginary part = 7.
c. In standard complex number form, 5i = 0 + 5i.
Real part = 0; imaginary part = 5.
d. In standard complex number form, 9 = 9 + 0i.
Real part = 9; imaginary part = 0.
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Operations with Complex Numbers
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Operations with Complex Numbers
Complex numbers can be added and subtracted easily by
adding or subtracting the real parts together and then
adding or subtracting the imaginary parts together. This is
similar to combining like terms with variables.
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Example 3 – Add or subtract complex numbers
Add or subtract the following complex numbers.
a. (2 + 8i) + (6 + 7i)
b. (5 – 4i) + (7 + 6i)
c. (6 + 3i) – (4 + 8i)
d. (2.5 + 3.8i) – (4.6 – 7.2i)
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Example 3 – Solution
a. (2 + 8i) + (6 + 7i) = 2 + 6 + 8i + 7i
= (2 + 6) + (8i + 7i)
Add the real parts.
Add the imaginary parts.
= 8 + 15i
b. 12 + 2i
c. (6 + 3i) – (4 + 8i) = 6 + 3i – 4 – 8i
Distribute the negative sign.
= (6 – 4) + (3i – 8i)
Subtract the real parts.
= 2 – 5i
Subtract the imaginary parts.
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Example 3 – Solution
cont’d
d. (2.5 + 3.8i) – (4.6 – 7.2i)
= (2.5 + 3.8i – 4.6 + 7.2i)
Distribute the negative sign.
= (2.5 – 4.6) + (3.8i + 7.2i)
Combine the real parts.
= –2.1 + 11i
Combine the imaginary parts.
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Operations with Complex Numbers
Because i =
, other powers of i can be calculated by
considering the following pattern,
i 5 = 1i = i
i=
i2 =
= –1
i3 =
=
i 6 = –1
= –i
i 4 = i 2i 2 = (–1)(–1) = 1
i 7 = –i
i8 = 1
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Operations with Complex Numbers
In most problems we will deal with, the most important
power of i that you should know and use is i 2 = –1.
When you multiply complex numbers, using this fact will
help you to reduce the answers to complex form.
Whenever you see i 2 in a calculation, you should replace it
with a – 1 and continue to combine like terms and simplify.
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Operations with Complex Numbers
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Example 4 – Multiplying complex numbers
Multiply the following complex numbers.
a. 3(4 + 9i)
b. 2i(7 – 3i)
c. (2 + 5i)(4 + 8i)
d. (3 + 2i)(3 – 2i)
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Example 4 – Solution
In all of these problems, we will use the distributive
property and simplify where possible.
a. 3(4 + 9i) = 12 + 27i
Distribute the 3.
b. 2i(7 – 3i) = 14i – 6i 2
Distribute the 2i.
= 14i – 6(–1)
Replace i 2 with –1.
= 14i + 6
Simplify.
= 6 + 14i
Put in standard complex form a + bi.
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Example 4 – Solution
c. (2 + 5i)(4 + 8i) = 8 + 16i + 20i + 40i 2
= 8 + 36i + 40i 2
= 8 + 36i + 40(–1)
= 8 + 36i – 40
cont’d
Use the distributive
property (FOIL).
Combine like terms.
Replace i 2 with –1.
Simplify.
= –32 + 36i
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Example 4 – Solution
d. (3 + 2i)(3 – 2i) = 9 – 6i + 6i – 4i 2
cont’d
Use the distributive
property (FOIL).
= 9 – 4i 2
Combine like terms.
= 9 – 4(–1)
Replace i 2 with –1.
=9+4
Simplify.
= 13
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Operations with Complex Numbers
Part d of Example 4 is an example of multiplying two
complex numbers that have a very special relationship.
These two complex numbers are what are called complex
conjugates of one another.
When complex conjugates are multiplied together, note that
the product is a real number.
Therefore, it has no imaginary part remaining.
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Operations with Complex Numbers
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Example 6 – Multiplying by complex conjugates
Multiply the following complex numbers by their conjugates.
a. 2 + 9i
b. 4 – 6i
c. 3.4i
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Example 6(a) – Solution
(2 + 9i)(2 – 9i) = 4 – 18i + 18i – 81i 2
Use the distributive property.
= 4 – 81i 2
Combine like terms.
= 4 – 81(–1)
Replace i 2 with –1.
= 4 + 81
Simplify.
= 85
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Example 6(b) – Solution
(4 – 6i)(4 + 6i) = 16 + 24i – 24i – 36i 2
cont’d
Use the distributive property.
= 16 – 36i 2
Combine like terms.
= 16 – 36(–1)
Replace i 2 with –1.
= 16 + 36
Simplify.
= 52
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Example 6(c) – Solution
3.4i (–3.4i) = –11.56i 2
cont’d
Distribute.
= –11.56(–1)
Replace i 2 with –1.
= 11.56
Simplify.
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Operations with Complex Numbers
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Example 7 – Dividing complex numbers
Divide the following. Give answers in the standard form for
a complex number.
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Example 7(a) – Solution
Since the denominator does not have an imaginary part,
we use the standard division algorithm.
Reduce the fraction and put it into the
standard form of a complex number.
Simplify.
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Example 7(b) – Solution
cont’d
Multiply the numerator and denominator by
the conjugate of the denominator. Use the
distributive property (FOIL).
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Example 7(b) – Solution
cont’d
Write in the standard form of a complex number.
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Example 7(c) – Solution
cont’d
Multiply the numerator and denominator by
the conjugate of the denominator. Use the
distributive property (FOIL).
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Example 7(c) – Solution
cont’d
Write in the standard form of a complex number.
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Example 7(d) – Solution
cont’d
Multiply the numerator and denominator by i.
Because the denominator does not have a real
part, multiply by i to rationalize the denominator.
Write in the standard form of a complex number.
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Solving Equations with Complex
Solutions
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Solving Equations with Complex Solutions
Some equations will have complex solutions. The most
common place in which we will see these types of solutions
is in working with quadratics.
The quadratic formula is a great tool to find both real and
complex solutions to any quadratic equation.
Now when a discriminant (b2 – 4ac) is a negative number,
we can write our solutions using complex numbers instead
of just saying that there are no real solutions.
This results in a more complete answer to the equation.
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Solving Equations with Complex Solutions
Notice that if a complex number is a solution to a
polynomial, the complex conjugate will also be a solution to
that equation.
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Example 8 – Solving equations with complex solutions
Solve the following equations. Give answers in the
standard form for a complex number.
a. t2 + 2t + 5 = 0
b. x2 = –25
c. x2 + 4x = –30
d. x3 – 10x2 + 29x = 0
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Example 8(a) – Solution
Use the quadratic formula.
The discriminant is –16, so the
answer will be a complex number.
Write in standard form.
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Example 8(b) – Solution
cont’d
Use the square root property.
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Example 8(c) – Solution
cont’d
Use the quadratic formula.
The discriminant is –104, so the
answer will be a complex number.
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Example 8(c) – Solution
cont’d
Write in standard form.
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Example 8(d) – Solution
cont’d
Factor the common term out.
Set each factor equal to zero
and continue to solve.
Use the quadratic formula.
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Example 8(d) – Solution
cont’d
The discriminant is –16, so the
answer will be a complex number.
Write in standard form.
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