Download Investigation: Complex Arithmetic

Investigation: Complex Numbers
Not all numbers are numbers that exist in our everyday world. These numbers
are called imaginary numbers.
 1  ______
All numbers are considered complex numbers, whether they are imaginary, real,
or both. Complex numbers can be written in the for a + bi
Part 1: Simplify the following complex numbers
a.  25
b.  19
7
d.
4
e.
g.
9 6
h.   18  7
Part 2: Types of numbers
Step 1: Family tree of numbers:
c.
 45
f.
 12
i. 24   75
Step 2: Classify each number
a. 0
b. 4.453423…
c. 3+4i
d. 4.453476
e. 3  7
f.  4
g. -5.4
h.
5
3
i.
4
Part 2: Graphing complex numbers
The x-axis is now the real number, the y-axis represents the imaginary.
Graph each number on the complex plane on the left. Then name each
complex number on the complex plane on the right.
a. 3+4i
b. -2-i
c. 3
d. -1+5i
Part 3: Find the value of each expression. Remember i   1
i = 1
i2 =
i3 =
i4 =
a. i24 =
b. i35 =
c. i50 =
d. i13 =
e. i102 =
f. i37 =
Investigation: Complex Arithmetic
Part 1: Add these complex numbers.
a. (2 – 4i) + (3 + 5i)
b. (7 + 2i) + (-2 + i)
c. (2 – 4i) – (3 + 5i)
d. (4 – 4i) – (1 – 3i)
Part 2: Now multiply these binomials. Express your products in the form a + bi.
Remember what is i2?
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)
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.
a.
7  2i
1 i
b.
10  11i
4  6i
c.
2i
8  6i
d.
2  4i
2  4i
e.
 1  2i
5i
f.
7i
3i
Part 5 Solve each equation.
a. x2 = -36
b. x2 = -28
c. -(x – 3)2 = 25
d. (2x + 7)2 – 15 = -28
e. 4(x – 11)2 + 27 = 3
e. -5(5x – 1)2 = 18