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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