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Section 4.8 – Complex Numbers
Students will be able to:
• To identify, graph, and perform operations with complex numbers
•To find complex number solutions of quadratic equations
Lesson Vocabulary:
imaginary unit imaginary number
complex number pure imaginary number
complex number plane abs val of comp #
complex conjugates
Section 4.8 – Complex Numbers
Essential Understanding:
The complex numbers are based on a number whose square is -1.
The imaginary unit “i” is the complex number whose square is -1. So i2 = -1, and i =
1
Section 4.8 – Complex Numbers
Section 4.8 – Complex Numbers
Problem 1:
How do you write by using the imaginary unit i?
18
Section 4.8 – Complex Numbers
Problem 1b:
How do you write by using the imaginary unit i?
12
Section 4.8 – Complex Numbers
Problem 1c:
How do you write by using the imaginary unit i?
25
Section 4.8 – Complex Numbers
Problem 1e:
Explain why ?
6464
Section 4.8 – Complex Numbers
An imaginary number is any number of the form
“a + bi”, where a and b are real number and b cannot equal 0.
Imaginary numbers and real numbers together make up the set of complex numbers.
Section 4.8 – Complex Numbers
Section 4.8 – Complex Numbers
In the complex number plane, the point (a, b) represents the complex number a + bi. To graph
a complex number, locate the real part on the horizontal axis and the imaginary part on the
vertical axis.
Section 4.8 – Complex Numbers
The absolute value of a complex number is its distance from the origin in the complex plane.
Section 4.8 – Complex Numbers
Problem 2:
What are the graph and absolute
value of each number?
a. -5 + 3i
b. 6i
Section 4.8 – Complex Numbers
Problem 2b:
What are the graph and absolute
value of each number?
a. 5 – i
b. 1 + 4i
Section 4.8 – Complex Numbers
To add or subtract complex numbers, combine the real parts and the imaginary parts separately.
If the sum of two complex numbers is 0, or 0 + 0i, then each number is the opposite, or additive
inverse, of the other.
The associative and commutative properties apply to complex numbers as well.
Section 4.8 – Complex Numbers
Problem 3:
What is each sum or difference?
a. (4 – 3i) + (-4 + 3i)
b. (5 – 3i) – (-2 + 4i)
c. (7 – 2i) + (-3 + i)
Section 4.8 – Complex Numbers
You multiply complex numbers a + bi and c + di as you would multiply binomials.
Problem 4:
What is each product?
a. (3i)(-5 + 2i)
b. (4 + 3i)(-1 – 2i)
c. (-6 + i)(-6 – i)
Section 4.8 – Complex Numbers
You multiply complex numbers a + bi and c + di as you would multiply binomials.
Problem 4b:
What is each product?
a. (3i)(7i)
b. (2 – 3i)(4 + 5i)
c. (-4 + 5i)(-4 – 5i)
Section 4.8 – Complex Numbers
Number pairs of the form a + bi and a – bi are complex conjugates. The product of complex
conjugates is a real number.
(a + bi)(a – bi) =
You can use complex conjugates to simplify quotients of complex numbers.
Section 4.8 – Complex Numbers
Problem 5:
What is each quotient?
a. b. 9 12
3
i
i
2 3
1 4
i
i
Section 4.8 – Complex Numbers
Problem 5:
What is each quotient?
a. b. 5 2
3 4
i
i
4
6
i
i
Section 4.8 – Complex Numbers
Problem 5:
What is the quotient?
8 7
8 8
i
i
Section 4.8 – Complex Numbers
Problem 6:
What is the factored form of 2x2 + 32?
What is the factored form of 5x2 + 20?
Section 4.8 – Complex Numbers
Problem 7:
What are the solutions of 2x2 – 3x +5 = 0?
What are the solutions of 3x2 – x + 2 = 0?
Section 4.8 – Complex Numbers
Section 4.8 – Complex Numbers
