Imaginary Numbers

Complex Numbers

a + bi​

​Let's Review. . .

Multiple Choice

Simplify: $$\sqrt{225}$$

• 15

• 20

• 25

• 112.5

Multiple Choice

Simplify: $$\sqrt{9}$$  

• 3

• 4.5

• 7

• 6

Multiple Choice

Simplify: $$\sqrt{64}$$

• 3

• 8

• 32

• 6

Multiple Choice

Solve:

x² = 100

• $$\pm10$$  

• $$\pm50$$  

• 10

• 50

Multiple Choice

Solve:

x² - 169 = 0

• $$\pm84.5$$  

• $$-13$$  

• $$\pm13$$  

• 84.5

​A Whole New World . . .

What are imaginary numbers?

Multiple Choice

What letter usually represent imaginary number?

• a

• i

• j

• x

Multiple Choice

• i

• -1

• 1

• -i

Steps:​

 

​

​Examples:

Multiple Choice

Your Turn!

$$\sqrt{-9}$$  

• 3

• 3i

• 9i

• $$3\sqrt[]{i}$$  

Multiple Choice

Simplify

• 6

• 6i

• -6

• i√6

Multiple Choice

$$x^2+81=0$$  

• $$\pm\sqrt{9i}$$  

• $$\pm9$$  

• $$\pm9i$$  

• $$\pm3i$$  

​

Now let's look at the powers of i

 

​

How to Simplify Powers of i:​​

​1) Divide the exponent by 4.

2) The remainder is the power of i​ the

expression is equivalent to. No

remainder means the expression is

equal to 1.​

​Examples:

​1) i¹³ 13/4 = 3 r 1 so i¹³ = i¹ = i

​

2) i²⁷ 27/4 = 6 r 3 so i²⁷ = i³ = - i​

​

​3) i⁸² 82/4 = 20 r 2 so i⁸² = i² = -1

​

​

​

Multiple Choice

Your Turn!

$$i^{22}$$  

• $$\sqrt{-1}$$  

• $$-1$$  

• $$-i$$  

• 1

Multiple Choice

$$i^{79}$$  

• i

• 1

• -1

• -i

Multiple Choice

$$i^7$$  

• i

• -i

• 1

• -1

​Adding Complex Numbers​

1) Add the real parts (a)

2) Add the imaginary parts (bi)​

​

Multiple Choice

(2 + 3i) + (6 - 5i)

• 5 + -i

• 8 - 2i

• 12 -15i

• 8 + 2i

Multiple Choice

(-10 + i) + (9 + 7i)

• -1 + 8i

• -1 + 7i

• 1 - 7i

• 1 + 8i