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4.4 Complex Numbers

Explore the world of complex numbers - imaginary and pure imaginary numbers, operations, properties, equations, and applications in electric circuits. Learn to simplify, add, subtract, multiply, and divide complex numbers.

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4.4 Complex Numbers

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  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 = • Numbers such as 6i, -4i, or iare called pure imaginary numbers. They represent square roots of negative numbers. Ie = ·2i

  3. Simplify Radicals A. B.

  4. Equations with Pure Imaginary Solutions • Solve 5y2 + 20 = 0.

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

  6. Properties with Imaginary Numbers • Commutative Property : a + b = b + a • Associative Property : a + (b + c) = (a + b) + c • Powers of i

  7. Day 2: Complex Numbers

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

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

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

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

  12. Conjugates • Find the product of (8 + 2i) and (8 – 2i) • Find the product of (3 – 5i) and (3 + 5i)

  13. Dividing Complex Numbers

  14. 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

  15. Application • ELECTRICITYIn 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.

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

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