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Unformatted text preview: ing in polar form: a) −7 + 7i b) 3 + 3i c) −i 28. Express each of the following in cartesian form: a) 2eπi/4 b) e−πi c) 3e2πi/3 d) 1 + √ 3i e) 1 − √ 3i f) −2 − 2i f) 9e−πi/4 d) 7eπi/6 e) 8e29πi 29. a) On the Argand diagram, choose a point z with positive real and imaginary parts and satisfying |z | = 2. Then indicate the positions of z and z −1 . b) Explain in simple English how you come to your conclusions. c) What is the distance between z −1 and the origin 0?
Chapter 1 : The Number System page 17 of 19 First Year Calculus c W W L Chen, 1982, 2005 30. Suppose that the complex number z satisﬁes |z | = 1. Prove that z = z −1 . 31. Let z be a non-zero complex number. Explain why 0, z −1 and z lie in a straight line on the Argand plane. 32. Suppose that the complex number z1 is a cube root of unity and the complex number z2 is a 4-th root of unity. Let z = z1 z2 . Show that z is a 12-th of unity. 33. Use de Moivre’s theorem to show that for every real numbe...
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