Worksheet1Soln

# Worksheet1Soln - MATH 2271 Differential Equations for...

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MATH 2271 Differential Equations for Scientists and Engineers Winter 2009 March 25, 2009 Worksheet on complex numbers—partial solutions 1. Express each of ¯ z, | z | , 1 /z, z in the form x + iy , for the following complex numbers. (a) z = i (b) z = 1 + i (c) z = - 1 2 + i 3 2 This is a matter of applying the definitions, as well as the formula for the square roots derived in lecture: for z = a + ib, z = ± a + a 2 + 2 2 + i b 2 a + a 2 + 2 2 . For example the solution to (b) is as follows. For z = 1 + i , ¯ z = 1 - i , | z | = 2, 1 /z = ¯ z | z | 2 = 1 2 - i 1 2 , and z = ± 1 + 2 2 + i 1 2 + 2 2 . 2. Using the definitions of conjugation (i.e., ¯ z ) and complex multiplication, show that each of the following holds for all w, z C . (a) w + z = ¯ w + ¯ z (b) w · z = ¯ w · ¯ z (c) | wz | = | w |·| z | Here again the problem involves a straightforward application of the definitions. The key is to write w, z in the standard notation, and then compare left and right sides of each of the given identities (or one can transform one side into the other by 1

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algebraic manipulation). For example, to verify (c), write
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