The Complex System
Skill Practice
Complex and Conjugates
Part 1: Using Conjugates
1. An expression with a radical in the denominator is not simplified. So,
simplify these by using conjugates.
3
a.
b.
5  13
7 2
1  2
c.
1
6  11
d.
15
3 5
2. Find the conjugate of 2 5  3 2
3. Find the reciprocal of the conjugate of 2 5  3 2
4. Find the conjugate of the reciprocal of 2 5  3 2
Part 2: The Basics of Complex Numbers
5. Use the definition of i to rewrite each of the following expressions.
4
a.
b.
2
c.
18  25
6. Simplify each of the expressions below.
a.
 3i 
d.
 3i 6 

3
b.
2
g. 1  i 3

2
 2i   5i 
2
2  16
2
d.

c. 3 2   50
e.
60
15
f. 4i  2  i 
h.
5
3  4i
i.

6i 2
6i 2
Part 3: Complex Numbers, Explorations
7. Explain why i3 = - i? What does i4 equal? Describe how you would
evaluate in where n could be any integer.
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8. If f(x) = x2 + 7x – 9, calculate the values in parts (a) through (c)
below.
a. f(-3)
b. f(i)
c. f(-3 + i)
9. Is 5 + 2i a solution to x2 – 10x = -29? How can you be sure?
10. Find the roots of y = (x + 5)2 + 9. How many x-intercepts does the
graph of this function have?
11. In parts (a) through (d) below, look for patterns as you calculate the
sum and the product for each pair of complex numbers.
a.
c.
2  i and 2  i
4  i and 4  i
b.
3  5i and 3  5i
d.
1  i 3 and 1  i 3
12. Each of the four pairs of complex numbers in problem 11 could be the
roots of a quadratic function. Create quadratic equations for each pair of
complex numbers in parts (a) through (d) in question 11.
13. There are several ways in which one can find the answers to question
12. Describe at least 3 different procedures that could be used to write
quadratic equations given its roots. If your group needs further guidance,
look to the end of this document.
14. Consider the equations y = x2 and y = 2x – 5. Use what you know
about systems and quadratics to solve the system formed by the equations.
What could these solutions mean?
15. Draw a pair of axes. Make the x-axis represent the real part of a
complex number. Make the y-axis represent the imaginary part.
a.
Plot the following complex numbers on your complex axes.
3 + 4i, 3 – 4i, -3 + 4i, -3 – 4i, 5, -5, 5i, -5i, and the 4 complex
numbers represented by 4  3i .
b. What do you notice about your graph? How far away from (0, 0)
is each point?
16. On the real number line, the distance from 0 to a point on the line is
defined as the absolute value of the number. Similarly, in the complex
plane, the absolute value of a complex number is its distance from zero or
the origin (0, 0). In the previous problem, the absolute value of all of
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those complex numbers was 5. For each of the following questions, a
sketch in the complex plane will help in visualizing the result.
a.
What is the absolute value of -8 + 6i?
b. What is the absolute value of 7 – 2i?
c.
What is
4i ?
d. What is the absolute value of a + bi?
Guidance for Question 13.
Consider the following questions to help guide your group in coming up with multiple ways to
write quadratic equations given roots.
a. How can we reverse the process of solving?
b. How can we use that we know about factors and zeroes?
c. How are the solutions related to the standard form of the equation?
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