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Unformatted text preview: d)i. For the division rule, suppose that c + di = 0, so that c = 0 or d = 0, whence c2 + d2 = 0. If a + bi = x + y i, c + di where x, y ∈ R, then a + bi = (c + di)(x + y i) = (cx − dy ) + (cy + dx)i. It follows that a = cx − dy, b = cy + dx. This system of simultaneous linear equations has the unique solution x= so that a + bi ac + bd bc − ad =2 +2 i. c + di c + d2 c + d2 The special case a = 1 and b = 0 gives 1 c − di . =2 c + di c + d2 This can also be obtained by noting that (c + di)(c − di) = c2 + d2 , so that 1 c − di c − di . = =2 c + di (c + di)(c − di) c + d2 It is also useful to note that in has exactly four possible values, with i2 = −1, i3 = −i and i4 = 1. Definition. Suppose that z = x + y i, where x, y ∈ R. The real number x is called the real part of z , and denoted by x = Rez . The real number y is called the imaginary part of z , and denoted by y = Imz . The set C = {z = x + y i : x, y ∈ R} is called the set of all complex numbers.
Chapter 1 : The Number Sy...
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 Fall '08
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 Math, Calculus

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