Chapter 4: Polynomial and Rational Functions 4.5: Complex Numbers Essential Question: What are the two complex numbers that have a square of -1?

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

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

4.5: 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

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

4.5: Complex Numbers Powers of i Example #4: Powers of i i1 = i 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

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

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

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

Chapter 4: Polynomial and Rational Functions 4 Chapter 4: Polynomial and Rational Functions 4.5: Complex Numbers (Part 2) Essential Question: What are the two complex numbers that have a square of -1?

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

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

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

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