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Unformatted text preview: where a, b, c, d ∈ R. Show that a = c and b = d.
Chapter 1 : The Number System page 16 of 19 First Year Calculus c W W L Chen, 1982, 2005 15. For each of the following complex numbers z , ﬁnd real numbers x and y such that z = x + iy , then show the positions of z and z on the Argand diagram, and determine the modulus and the principal argument of z : 3 + 4i a) z = (1 + i)4 b) z = 1 − 2i 16. Let z = x + iy , where x, y ∈ R. a) Write down |z |2 and (Rez )2 . b) Hence prove that |z | ≥ Rez . c) For what values of z does equality hold? 17. Let z = 3 − 4i. a) Find z , |z | and z −1 . b) Verify that z −1 = z/|z |2 . 18. Solve each of the following equations and leave your answers in rectangular form: a) z 2 + 4z + 5 = 0 b) z 2 + iz − 1 = 0
3 3 19. Solve the equation z 2 + z + 1 = 0. If the solutions are z1 and z2 , calculate z1 and z2 . 20. Consider the equation z 2 − iz − 1 = 0. a) Solve the equation. b) Are your solutions conjugates of each other? c) Comment on the results. 21. Find the square roots of 5 + 1...
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