Download Section 4.8 Notes - Verona Public Schools

Section 4.8- Complex Numbers
Essential Question: What is an imaginary number?
Do Now:
Square Root of a Negative Real Number

Consists of √−1 = ______

𝑖2 =

For any positive number 𝑎, √−𝑎 = √−1 ∙ 𝑎 =_____________

Ex. √−5 =_______
NOTE: (√−5)2= ______
Example 1: Simplifying a Number using 𝑖
How do you write each number by using the imaginary unit 𝑖?
a. √−12
b. √−25
d. Explain why √−64 ≠ −√64.
c. √−7
Complex Number Plane

Use the point (a,b)

Real part is the _____________ axis

Imaginary part is the _____________ axis

Absolute value of a complex number is distance from origin in the complex plane
o |𝑎 + 𝑏𝑖| = √𝑎2 + 𝑏 2
Example 2- Graphing in the Complex Number Plane
What are the graph and absolute value of 5 − 𝑖?
Mathematical Operations Using Complex Numbers
Addition and Subtraction

Combine real parts and imaginary parts separately
o Similar to combining ______ ________
Example 3- Adding and Subtracting Complex Numbers
What is each sum or difference?
a. (7 − 2𝑖) + (−3 + 𝑖)
b. (1 + 5𝑖) − (3 − 2𝑖)
c. (8 + 6𝑖) − (8 − 6𝑖)
d. (−3 + 9𝑖) + (3 + 9𝑖)
Multiplication

Similar to FOIL or Distributive Property

Recall 𝑖 2 = _____
Example 4- Multiplying Complex Numbers
What is each product?
a. (7𝑖)(3𝑖)
b. (2 − 3𝑖)(4 + 5𝑖)
c. (−4 + 5𝑖)(−4 − 5𝑖)
Division

Use complex conjugates to simplify quotients
o _____________ _______________are complex conjugates
o To simplify, multiply both _____________ and ________________ by
complex conjugates

Use multiplication more than division
Example 5- Dividing Complex Numbers
What is each quotient?
a.
5−2𝑖
3+4𝑖
b.
4−𝑖
6𝑖
c.
8−7𝑖
8+7𝑖
Example 6- Finding Imaginary Solutions
Use the quadratic formula to find the solutions of each equation.
a. 3𝑥 2 − 𝑥 + 2 = 0
HW: p. 254 #45, 47-55, 61-63
b. 𝑥 2 − 4𝑥 + 5 = 0