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workout 33a

# workout 33a - HPHY 253 Workout 33a Complex Numbers Dr A E...

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HPHY 253 Workout 33a Complex Numbers Dr. A. E. Bak Name 1. Convert the following complex numbers to polar form. Limit the phase range to 0 2 ± . (a) z = 1 + i . We have j z j = q [Re ( z )] 2 + [Im ( z )] 2 = q (1) 2 + (1) 2 = p 2 and arg ( z ) = tan 1 Im ( z ) Re ( z ) ± = tan 1 ² 1 1 ³ = tan 1 (1) = ± 4 : Therefore, we obtain z = p 2 e + 4 as the sought result. (b) z = ± p 3 + i . We have j z j = q [Re ( z )] 2 + [Im ( z )] 2 = r ´ ± p 3 µ 2 + (1) 2 = 2 and arg ( z ) = tan 1 Im ( z ) Re ( z ) ± = tan 1 ² 1 ± p 3 ³ = 5 ± 6 : Therefore, we obtain z = 2 e +5 6 as the sought result. (c) z = 4 i . We have j z j = q [Re ( z )] 2 + [Im ( z )] 2 = q (0) 2 + (4) 2 = 4 and arg ( z ) = tan 1 Im ( z ) Re ( z ) ± = tan 1 ² 4 0 ³ = tan 1 (+ 1 ) = ± 2 : Therefore, we obtain z = 4 e + 2 as the sought result. (d) z = 1 = (1 + i ) . From part a , we have 1 + i = p 2 e + 4 ; and so z = 1 1 + i = 1 p 2 e + 4 = 1 p 2 e 4 : Although this expression is true, the phase

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workout 33a - HPHY 253 Workout 33a Complex Numbers Dr A E...

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