Chapter 4: Polynomial and Rational Functions 4.5: Complex Numbers

1. Chapter 4: Polynomial and Rational Functions4.5: Complex Numbers Essential Question: What are the two complex numbers that have a square of -1?

2. 4.5: Complex Numbers • Properties of the Complex Number System • The complex number system contains all real numbers • Addition, subtraction, multiplication, and division of complex numbers obey the same rules of arithmetic that hold in the real number system with one exception: • The exponent laws hold for integer exponents, but not necessarily for fractional ones • We don’t need to worry about this for now, I just needed to list the exception • The complex number system contains a number, denoted i, such that i2 = -1 • Every complex number can be written in the standard form: • a + bi • a + bi = c + di if and only if a = c and b = d • Numbers of the form bi, where b is a real number, are called imaginary numbers. Sums of real and imaginary numbers, numbers of the form a + bi, are called complex numbers

3. 4.5: Complex Numbers • Example #1: Equaling Two Complex Numbers • Find x and y if 2x – 3i = -6 + 4yi • The real number parts are going to be equal • 2x = -6 • x = -3 • The imaginary number parts are going to be equal • -3i = 4yi • -3/4 = y

4. 4.5: Complex Numbers • Example #2: Adding, Subtracting, and Multiplying Complex Numbers • (1 + i) + (3 – 7i) • Combine like terms • 4 – 6i • (4 + 3i) – (8 – 6i) • Distribute, then combine terms • 4 + 3i – 8 + 6i = -4 + 9i • 4i(2 + ½ i) • Distribute and simplify • 8i + 2i2 = 8i + 2(-1) = -2 + 8i • (2 + i)(3 – 4i) • FOIL and simplify • 6 – 8i + 3i – 4i2 = 6 – 8i + 3i – 4(-1) = 6 – 8i + 3i + 4 = 10 – 5i

5. 4.5: Complex Numbers • Example #3: Products and Powers of Complex Numbers • (3 + 2i)(3 – 2i) • FOIL • 9 – 6i + 6i – 4i2 = 9 – 4(-1) = 9 + 4 = 13 • (4 + i)2 • 16 + 4i + 4i + i2 = 16 + 4i + 4i + (-1) = 15 + 8i

6. 4.5: Complex Numbers • Powers of i • i1 = i • i2 = -1 • i3 = i2 • i = -1 • i = -i • i4 = i2 • i2 = -1 • -1 = 1 • i5 = i4 • i = 1 • i = i • And we keep repeating from there… • Example #4: Powers of i • Find i54 • The remainder when 54 / 4 is 2, so i54 = i2 = -1

7. 4.5: Complex Numbers • Complex Conjugates • The conjugate of the complex number a + bi is the number a – bi, and the conjugate of a – bi is a + bi • Conjugates multiplied together yield a2 + b2 • (a – bi)(a + bi) = a2 + abi – abi – b2i2 = a2 – b2(-1) = a2 + b2 • The conjugate is used to eliminate the i from the complex number, and is used to remove the use of i in the denominator of fractions

8. 4.5: Complex Numbers • Example #5: Quotients of Two Complex Numbers • Simplify • multiply top & bottom by the conjugate of the denominator

9. 4.5: Complex Numbers • Assignment • Page 300 • Problems 1-35 & 55-57, odd problems • Show work where necessary (e.g. FOILing, converting to i) • Due tomorrow

10. Chapter 4: Polynomial and Rational Functions4.5: Complex Numbers (Part 2) Essential Question: What are the two complex numbers that have a square of -1?

11. 4.5: Complex Numbers • Square Roots of Negative Numbers • Because i2 = -1, • In general, • Take the i out of the square root, then simplify from there • Example #6: Square Roots of Negative Numbers

12. 4.5: Complex Numbers • Complex Solutions to a Quadratic Equation • Find all solutions to 2x2 + x + 3 = 0

13. 4.5: Complex Numbers • Zeros of Unity • Find all solutions of x3 = 1 • Rewrite equation as x3 - 1 = 0 • Use graphing calculator to find the real roots (1) • Factor that out • (x – 1)(x2 + x + 1) = 0 • x = 1 or x2 + x + 1 = 0

14. 4.5: Complex Numbers • Assignment • Page 300 • Problems 37-71 (odd) (skip 55/57, you did that last night) • Due tomorrow • You must show work