5.4 Complex Numbers Goals p Perform operations with complex numbers. p Apply complex numbers to fractal geometry. Your Notes VOCABULARY Imaginary unit i The imaginary unit i is defined as i ϭ ͙Ϫෆ1. Complex number A number a ϩ bi where a and b are real numbers and i is the imaginary unit Standard form of a complex number The form a ϩ bi where a and b are real numbers and i is the imaginary unit. The number a is the real part of the complex number and bi is the imaginary part of the complex number. Imaginary number A complex number a ϩ bi where b 0 Pure imaginary number A complex number a ϩ bi where a ϭ 0 and b 0 Complex plane A coordinate plane where each point (a, b) represents a complex number a ϩ bi. The horizontal axis is the real axis and the vertical axis is the imaginary axis. Complex conjugates Two complex numbers of the form a ϩ bi and a Ϫ bi Absolute value of a complex number If z ϭ a ϩ bi, then the absolute value of z, denoted z, is a nonnegative real number defined as z ϭ ͙aෆ2 ϩ bෆ2 . Geometrically, the absolute value of a complex number is the number’s distance to the origin. 104 Algebra 2 Notetaking Guide • Chapter 5 Your Notes THE SQUARE ROOT OF A NEGATIVE NUMBER Property Example 1. If r is a positive real number, ͙ෆϪ5 i ͙5 then ͙Ϫෆr i ͙rෆ. 2. By Property (1), it follows (i ͙5ෆ)2 i2 p 5 Ϫ5 that (i ͙rෆ)2 r. Example 1 Solving a Quadratic Equation Solve 2x2 3 ϭϪ15. Solution 2x2 3 ϭϪ15 Write original equation. 2x2 ϭϪ18 Subtract 3 from each side. x2 ϭϪ9 Divide each side by 2 . x ϭϮ͙Ϫෆ9 Take square roots of each side. x ϭϮi͙9ෆ Write in terms of i. x ϭϮ3i Simplify the radical. The solutions are 3i and Ϫ3i . Example 2 Plotting Complex Numbers Plot the complex numbers in the complex plane. a. 1 ϩ i b. Ϫ2 Ϫ 2i c. 3 Ϫ 3i Solution a. To plot 1 ϩ i, start at the origin, imaginary move 1 unit to the right , and 1 ϩ i i then 1 unit up . Ϫ3 Ϫ1 1 real Ϫi b. To plot Ϫ2 Ϫ 2i, start at the origin, Ϫ2 2i move 2 units to the left , and 3 3i Ϫ3i then 2 units down . c. To plot 3 Ϫ 3i, start at the origin, move 3 units to the right , and then 3 units down . Lesson 5.4 • Algebra 2 Notetaking Guide 105 Your Notes Example 3 Adding and Subtracting Complex Numbers Write the expression (5 ؉ i) ؉ (1 ؊ 2i) as a complex number in standard form. (5 ϩ i) ϩ (1 Ϫ 2i) ϭ ( 5 ϩ 1 ) ϩ ( 1 Ϫ 2 )i Complex addition 6 Ϫ i Standard form Checkpoint Complete the following exercises. 1. Solve 5x2 ϩ 2 ϭϪ8. 2. In which quadrant of the complex plane is 1 Ϫ 3i? Ϯi ͙2ෆ Quadrant IV 3. Write 3 Ϫ (7 ϩ 8i) ϩ (5 Ϫ 6i) as a complex number in standard form. 1 Ϫ 14i Example 4 Multiplying and Dividing Complex Numbers Write the expression as a complex number in standard form. 6 Ϫ 4i a. (1 Ϫ 4i)(3 ϩ 5i) b. ᎏᎏ 1 ϩ i Solution a. (1 Ϫ 4i)(3 ϩ 5i) ϭ 3 ϩ 5i Ϫ 12i 20i2 Use FOIL. ϭ 3 Ϫ 7i 20(Ϫ1) Simplify and . Ϫ1 ؍ use i2 ϭ 23 7i Standard form 6 Ϫ 4i 6 Ϫ 4i 1 Ϫ i b. ᎏᎏ ᎏᎏ p ᎏᎏ Multiply by 1 Ϫ i , the ϩ ϩ Ϫ .i 1 i 1 i conjugate of 1 ؉ i 1 6 Ϫ 6i Ϫ 4i ϩ 4i2 ᎏᎏᎏ Use FOIL. 1 Ϫ i ϩ i Ϫ i2 2 Ϫ 10i ᎏᎏ Simplify. 2 ϭ 1 Ϫ 5i Write in standard form. 106 Algebra 2 Notetaking Guide • Chapter 5 Your Notes Checkpoint Write the expression in standard form. 3 ϩ 2i 4. (4 Ϫ 5i)(4 ϩ 5i) 5. ᎏᎏ 2 Ϫ i 4 7 41 ᎏᎏ ϩ ᎏᎏi 5 5 Example 5 Finding Absolute Values of Complex Numbers Find the absolute value of each complex number. Which number is farthest from the origin in the complex plane? a. Ϫ2 Ϫ 3i b. 2 ϩ i c. Ϫ1 ϩ 3i Solution Ϫ1 ϩ 3i imaginary 3i a. Ϫ2 Ϫ 3i z ͙____10ළළ 2 ϩ i ϭ ͙(ෆϪ2)2 ϩෆ(Ϫ3)ෆ2 Ϫ3 Ϫ1 1 real ϭ ͙ෆ ≈ 13 3.6 z ____͙ 13ළළ z ___͙ 5ළ Ϫ3i b. 2 ϩ i Ϫ2 Ϫ3i ϭ ͙(2)ෆ2 ϩෆ(1)2 ϭ ͙5ෆ ≈ 2.2 c. Ϫ1 ϩ 3i ϭ ͙(ෆϪ1)2 ϩෆ(3)2 ϭ ͙ෆ10 ≈ 3.2 Because Ϫ2 Ϫ 3i has the greatest absolute value, it is farthest from the origin in the complex plane. Lesson 5.4 • Algebra 2 Notetaking Guide 107 Your Notes COMPLEX NUMBERS IN THE MANDELBROT SET To determine whether a complex number c is in the Mandelbrot set, consider the function f(z) ϭ z2 ϩ c and this ϭ ϭ infinite list of complex numbers: z0 0, z1 f(z0), ϭ ϭ z2 f(z1), z3 f(z2), … • If the absolute values z0, z1, z2, z3, … are all less than some fixed number N, then c is in the Mandelbrot set. • If the absolute values z0, z1, z2, z3, … become infinitely large , then c is not in the Mandelbrot set. Example 6 Complex Numbers in the Mandelbrot Set Tell whether c ϭϪi belongs to the Mandelbrot set. Solution Let f(z) ϭ z2 Ϫ i. ϭ ϭ z0 0 z0 0 ϭ ϭ 2 Ϫ ϭϪ ϭ z1 f( 0 ) 0 i i z1 1 ϭ Ϫ ϭ Ϫ 2 ϭϪ ≈ z2 f( i ) ( i) i 1 i z2 1.41 ϭ Ϫ ϭ Ϫ 2 ϭ ≈ z3 f( 1 i ) ( 1 i) i i z3 1 ϭ ϭ 2 ϭϪ ≈ z4 f( i ) (i) i 1 i z4 1.41 The absolute values alternate between 1 and ͙2ෆ , so all the absolute values are less than N ϭ 2 . Therefore, c ϭϪi belongs to the Mandelbrot set. Checkpoint Complete the following exercises. 6. Find the absolute value 7. Tell whether c ϭ 2 belongs of 8 Ϫ 6i. to the Mandelbrot set. Homework 10 no 108 Algebra 2 Notetaking Guide • Chapter 5.