634 CHAPTER 9 Polar Coordinates; Vectors of arguments is π π ⋅ = 1 9 6 2 3 ; this means that the three points are equally spaced around the circle, as shown in Figure 44. These results are not coincidental. You are asked to show that these results hold for complex n th roots in Problems 67–69. Figure 44 Imaginary axis Real axis 1 21 22 1 21 22 2 2 z0 O 3 2)2 x2 1 y2 5 ( 2p –– 9 2p –– 3 2p –– 3 2p –– 3 z1 z2 Now Work PROBLEM 57 Historical Feature The Babylonians, Greeks, and Arabs considered square roots of negative quantities to be impossible and equations with complex solutions to be unsolvable. The first hint that there was some connection between real solutions of equations and complex numbers came when Girolamo Cardano (1501—1576) and Tartaglia (1499—1557) found real roots of cubic equations by taking cube roots of complex quantities. For centuries thereafter, mathematicians worked with complex numbers without much belief in their actual existence. In 1673, John Wallis appears to have been the first to suggest the graphical representation of complex numbers, a truly significant idea that was not pursued further until about 1800. Several people, including Karl Friedrich Gauss (1777—1855), then rediscovered the idea, and graphical representation helped to establish complex numbers as equal members of the number family. In practical applications, complex numbers have found their greatest uses in the study of alternating current, where they are a commonplace tool, and in the field of subatomic physics. 1. The quadratic formula works perfectly well if the coefficients are complex numbers. Solve the following. (a) ( ) − + − + = z i z i 2 5 3 5 0 2 (b) ( ) − + − − = z i z i 1 2 0 2 Historical Problems John Wallis ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 9.3 Assess Your Understanding 1. The conjugate of − − i 4 3 is . (p. A60) 2. The Sum Formulas for the sine and cosine functions are: (a) ( ) + = A B sin . (p. 514) (b) A B cos ( ) + = . (p. 511) 3. π = sin 2 3 ; π = cos 4 3 (pp. 403–405) 4. Simplify: ⋅ = e e 2 5 ; ( ) = e4 3 (pp. A7–A9) Concepts and Vocabulary 5. In the complex plane, the x -axis is referred to as the axis, and the y -axis is called the axis. 6. When a complex number z is written in the polar form θ θ ( ) = + z r i cos sin , the nonnegative number r is the or of z, and the angle θ θ π ≤ < , 0 2 , is the of z. 7. Suppose = θ z r ei 1 1 1 and = θ z r ei 2 2 2 are two complex numbers. Then = z z1 2 8. True or False If = θ z rei is a complex number and n is an integer, then = θ z r e . n n i 9. Every nonzero complex number has exactly distinct complex cube roots. 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure Credit: Science & Society Picture Library/Contributor/ Getty Images
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