Download Investigation: Complex Arithmetic

Investigation: Applying Complex Numbers
When working with complex numbers, the rules are similar to those you use when working with real
numbers.
Part 1: Add these complex numbers. (Hint: it’s just like adding like terms)
a. (2 – 4i) + (3 + 5i)
b. (7 + 2i) + (-2 + i)
c. (2 – 4i) – (3 + 4i)
d. (4 -4i) – (1 – 3i)
Part 2: Now multiply these binomials. Express your products in the form a + bi. (Hint: Use FOIL, use
the fact that i2 = -1, and then combine like terms)
a. (2 – 4i)(3 + 5i)
b. (7 + 2i)(-2 + i)
c. (2 – 4i)2
d. (4 – 4i)(1 -3i)
Part 3: The conjugate of a + bi is a – bi. Let’s see what happens when we add or subtract them
together.
a. (2 – 4i) + (2 + 4i)
b. (7 + 2i) + (7 – 2i)
c. (2 – 4i)(2 + 4i)
d. (-4 + 4i)(-4 – 4i)
State what happens when we multiply (or add) with conjugates: ______________________________
Part 4: Recall rationalizing the denominator with radicals
3
2 3 2


2
2 2
We will use a similar technique to change the complex denominator to a real number by using
conjugates. Once you have a real number in the denominator, divide to get an answer in the form a +
bi. (Hint: Multiply the conjugate of the denominator to the top and bottom part of the fraction. FOIL
the top and bottom and simplify. Then simplify the fraction itself.)
a.
7  2i
1 i
b.
10  11i
4  6i
c.
2i
8  6i
d.
2  4i
2  4i
Practice 7-5
1. Add or subtract
a. (5 – 1i) + (3 + 5i)
c.
(2 + 3i) + (2 – 5i)
2. Multiply.
a. (5 – 1i)(3 + 5i)
c.
3i(2 – 5i)
b. (6 + 2i) – (-1 + 2i)
d. (2.35 + 2.71i) – (4.91 + 3.32i)
b. 6(-1 + 2i)
d. (2.35 + 2.71i)(4.91 + 3.32i)
3. Find the conjugate of each complex number.
a. 5 – i
b. -1 + 2i
c. 2 + 3i
d. -2.35 – 2.71i
4. Draw Venn diagrams to show the relationships between these sets of numbers.
a. Real number and complex numbers.
b. Rational numbers and irrational numbers
c. Imaginary numbers and complex numbers
d. Imaginary numbers and real numbers
e. Complex numbers, real numbers, and imaginary numbers
5. Rewrite this quadratic equation in general form.
[x – (2+i)][x – (2 – i)] = 0
6. Rewrite the quotient
2  3i
in the form a + bi.
2i
7. Solve each equation. Use substitution to check your solutions. Label each solution as real,
imaginary, and/or complex.
a. x2 – 1 =0
b. x2 + 1 = 0
c. x2 – 4x + 6 = 0
d. x2 + x = -1
e. -2x2 + 4x = 3
Name: _______________________
Homework 7.5 Part 2 Complex Arithmetic
Date: ___________
1. Simplify the following. Put your answer in the form a + bi
2
a. 2.3   54
b.  3  9i 
c. (-5 + 6i) – (1 – i)
d. (-2.4 – 5.6i) + 5.9 + 1.8i)
e. (2.5 + 1.5i)(3.4 – 0.6i)
g. -4i(-6 + i)
h. (2+3i)(4 – 5i)
l. i16
m.
2
3i
i. i30
n.
2. Find the conjugate of each complex number.
a. 5 -4i
b. 7i
4i
5  2i
j. i9
f.
1 i
1 i
k. i22
o.
c. -3.25 + 4.82i
3  5i
6i
3. Solve the equations. Label each solution as real, imaginary, and/or complex.
a. x2 – 2x + 5 = 0
b. x2 + 7 = 0
c. x(x – 5)= 1
d. x2 + x + 1 = 0
e. 4x2 + 9 = 0
f. (x + 7)(x – 3) = 5 – 2x
4. Write a quadratic function in general form that has the given zeros and leading coefficient of 1 (a =
1).
a. x = -4, x = 7
b. x = 11i, x = -11i
c. x = -2 + 3i, x = -2-3i
5. Name the complex number associated with each point A through C, on the complex plane shown.