4.4 Complex Numbers.

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Presentation transcript:

4.4 Complex Numbers

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

Simplify Radicals A. B.

Equations with Pure Imaginary Solutions Solve 5y 2 + 20 = 0.

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

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 =

Day 2: Complex Numbers

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

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

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

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

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

Dividing Complex Numbers

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

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.

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