1. 4.4 Complex Numbers

2. New kinds of numbers
In previous classes, you have heard “all real numbers” What are unreal numbers?
The “imaginary Unit” i is defined to be i2 = -1
Or i = −1
Numbers such as 6i, -4i, or i 3 are called pure imaginary numbers. They represent square roots of negative numbers. Ie −4 = 4 · −1 = 2i

3. Simplify Radicals A. B.

4. Equations with Pure Imaginary Solutions
Solve 5y = 0.

5. Operations with Pure Imaginary Numbers
A. Simplify –3i ● 2i. B.

6. Properties with Imaginary Numbers
Commutative Property : a + b = b + a
Associative Property : a + (b + c) = (a + b) + c
Powers of i
i1 = i
i2 = -1
i3 = i2 · i = -i
i4 = i2· i2 = 1
i5 = i4 ·i = i
i6 = i4 ·i2= -1
i7 =
i8 =

7. Day 2: Complex Numbers

9. Equate Complex Numbers
Find the values of x and y that make the equation 2x + yi = –14 – 3i true.

10. Operations with complex Numbers: Add/Subtract
Combine Real parts, combine imaginary parts
A. Simplify (3 + 5i) + (2 – 4i).
B. Simplify (4 – 6i) – (3 – 7i).

11. Operations with complex Numbers: Multiply
Multiply using distributive property (FOIL)
(5 + 2i)(4 – 6i)
(8 – 3i)2

12. Conjugates a + bi and a – bi
When you multiply conjugates together all imaginary numbers go away.

13. Conjugates Find the product of (8 + 2i) and (8 – 2i)

14. Dividing Complex Numbers

15. Writing Equations
Write a quadratic equation in standard form with 5i and -5i as solutions.
Write a quadratic equation in standard form with 3+i and 3 – i

16. Application
ELECTRICITY In an AC circuit, the voltage E, current I, and impedance Z are related by the formula E = I ● Z Find the voltage in a circuit with current 1 + 4j amps and impedance 3 – 6j ohms.

17. Find the values of x and y that make the equation 3x – yi = 15 + 2i true.
A. x = 15 y = 2
B. x = 5 y = 2
C. x = 15 y = –2
D. x = 5 y = –2