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1.6 Perform Operations with Complex Numbers p. 41 What is an imaginary number? How is it defined? What is a complex number? How is it graphed? How do you add, subtract, multiply and divide complex numbers? What is a complex conjugate? When do you use it? How do you find the absolute value of a complex number? Define Imaginay •She used her imagination to write the story. •His imagination helped him with the role in the play. •He was able to draw the picture of the house he wanted to build using his imagination. •She had a tea party with her imaginary friends. •Define imagination. •Define imaginary. •What is the difference? Solve: 2 x =1 2 x = -1 Not all quads. have real number solutions. Imaginary numbers get around this problem. Imaginary Unit Definition i = √−1 and i 2 = −1 Practice: Solve for x 2 2 x = -13 x = -38 x2 + 11 = 3 x2 -8 = -36 2 3x -7 = -31 2 5x + 33 = 3 Define the word: complex • He gave a complex explanation. • It was a complex math problem. • He turned a simple solution into a complex solution. Rube Goldberg Machine Complex number A complex number is made up of a real number and an imaginary number. Standard form for a complex number is: a + bi a is the real part and bi is the imaginary part. If a = 0 and b ≠ 0, then a + bi is a pure imaginary number (0 + bi). Adding and Subtracting Complex Numbers (a + bi) + (c + di) = (a + c) + (b + d)i (4 – i) + (3 + 2i) (7 – 5i) – (1 – 5i) 6 – (−2 + 9i) + (−8 + 4i) Multiplying Complex Numbers (a + bi)(c + di) = (ac - bd) + (ad + bc)i 5i(−2 + i) (7 – 4i)(−1 + 2i) (6 + 3i)(6 – 3i) Homework! Hw 1.6a pg 45-46 #4-20 even, 22-27 all 1.6 Day 2 Menu Divide Complex Numbers Plot Complex Numbers Absolute Values of Complex Numbers Complex Conjugate Two complex numbers of the form a + bi and a - bi are called Complex Conjugates. The product of complex conjugates is always real. (a+bi)(a-bi) = a2 + b2 Dividing Complex Numbers Write the quotient in standard form. 1. Multiply by complex conjugate 2. FOIL 3. Use i 2 = 1 & simplify 5 + 3i 1− 2i 7+5i 1-4i Plotting Complex Numbers a)2 – 3i b) -3 + 2i c) 4i Imaginary Axis Real Axis Finding Absolute Values of Complex Numbers The absolute value of a complex number z = a + bi is denoted as |z| is a non-negative real number defined as |z| = √a2 + b2 Fine the absolute value of -4+3i Finding Absolute Values of Complex Numbers • MORE PRACTICE!!!! Plot each point in the complex plane. Then find the absolute value for each complex number. 1. 4-i 2. -3-4i 3. 2+5i 4. -4i Assignment 1.6b p. 46, 30-33 all, 34-48 evens, 51-56 all