# ans1 - Math 185 Sample Answers to Problem Set#1 Page 4 2...

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Unformatted text preview: Math 185. Sample Answers to Problem Set #1 Page 4 2. Write z = x + iy . Then iz = ix + i 2 y =- y + ix , so Re( iz ) =- y =- Im z and Im( iz ) = x = Re z . 4. If z = 1 + i then z 2- 2 z +2 = (1+ i ) 2- 2(1+ i )+2 = (1+2 i + i 2 )- 2- 2 i +2 = (1- 1- 2+2)+(2 i- 2 i ) = 0 and if z = 1- i then z 2- 2 z +2 = (1- i ) 2- 2(1- i )+2 = (1- 2 i + i 2 )- 2+2 i +2 = (1- 1- 2+2)- 2 i +2 i = 0 . Page 7 2. We will use the (x,y) notation for this problem. Write z = ( x, y ) . a .- z is defined by (5) on page 3, so- z = (- x,- y ) . Also,- 1 = (- 1 , 0) , so by (4) on page 2, (- 1) z = (- 1 , 0)( x, y ) = ((- 1) x- y, x + (- 1) y ) = (- x,- y ) =- z . b . By (8) on page 4, 1 1 /z = x x 2 + y 2 ,- y x 2 + y 2- 1 = x x 2 + y 2 ( x x 2 + y 2 ) 2 + (- y x 2 + y 2 ) 2 ,-- y x 2 + y 2 ( x x 2 + y 2 ) 2 + (- y x 2 + y 2 ) 2 = x ( x 2 + y 2 ) x 2 + (- y ) 2 , y ( x 2 + y 2 ) x 2 + (- y ) 2 = ( x, y ) = z ....
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## This note was uploaded on 09/01/2010 for the course MATH 185 taught by Professor Lim during the Fall '07 term at Berkeley.

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ans1 - Math 185 Sample Answers to Problem Set#1 Page 4 2...

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