Chap 1 The Number System

And zz x y i x y i y 2i 2i proof write z x

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Unformatted text preview: stem page 7 of 19 ac + bd c2 + d2 and y= bc − ad , c2 + d2 First Year Calculus c W W L Chen, 1982, 2005 Example 1.5.1. We have (1 + 2i)2 −3 + 4i (−3 + 4i)(1 + i) −7 + i 71 = = = = − + i. 1−i 1−i (1 − i)(1 + i) 2 22 Hence Re (1 + 2i)2 7 =− 1−i 2 and Im (1 + 2i)2 1 =. 1−i 2 Example 1.5.2. We have 1 + i + i2 + i3 = 0 and 5 + 7i2003 = 5 − 7i. Definition. Suppose that z = x + y i, where x, y ∈ R. Then the complex number z = x − y i is called the conjugate of the complex number z . PROPOSITION 1B. Suppose that z ∈ C. Then Rez = z+z 2 and Imz = z−z . 2i Proof. Write z = x + y i, where x, y ∈ R. Then z+z (x + y i) + (x − y i) = =x 2 2 as required. PROPOSITION 1C. Suppose that z, w ∈ C. Then z+w =z+w and zw = z w. and z−z (x + y i) − (x − y i) = =y 2i 2i Proof. Write z = x + y i and w = u + v i, where x, y, u, v ∈ R. Then z + w = (x + u) + (y + v )i = (x + u) − (y + v )i = (x − y i) + (u − v i) = z + w and zw = (x + y i)(u + v i) = (xu − yv ) + (xv + yu)i = (xu − yv ) − (xv + yu)i = (x − y i)(u − v i) = z w as re...
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This note was uploaded on 02/01/2009 for the course MATH 2343124 taught by Professor Staff during the Fall '08 term at UCSD.

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