Explanation Given Information: The complex numbers in polar form, z 1 = 2 + 2 i z 2 = 1 + i Formula used: For two complex numbers, given by, z 1 = r 1 ( cos θ 1 + i sin θ 1 ) z 2 = r 2 ( cos θ 2 + i sin θ 2 ) The division of two complex numbers is calculated as, z 1 z 2 = r 1 r 2 ( cos ( θ 1 − θ 2 ) + i sin ( θ 1 − θ 2 ) ) Calculation: Consider the given complex number to write it in the polar form, z 1 = 2 + 2 i z 2 = 1 + i (I) For a complex number z = x + i y , the polar form is given by, z = r ( cos θ + i sin θ ) (II) Here, x = r cos θ and y = r sin θ , dividing the value of y by x , to get tan θ = y x (III) Also, the value of r is called as moduli of the complex number, given by, For z 1 = 2 + 2 i r = x 2 + y 2 r = ( 2 ) 2 + ( 2 ) 2 r = 4 + 4 r = 2 2 r = 2 2 (IV) For any complex number z = x + i y , the sign of the value of x and y determine that in which quadrant the value of z = x + i y would lie, If the value of x lies is positive and the value of y is positive, then the angle θ lies in the first quadrant having the value of θ to be θ (V) Also, if the value of x lies is negative and the value of y is positive, then the angle θ lies in the second quadrant having the value of θ to be π − θ (VI) Also, if the value of x lies is negative and the value of y is negative, then the angle θ lies in the third quadrant having the value of θ to be π + θ (VII) Also, if the value of x lies is positive and the value of y is negative, then the angle θ lies in the fourth quadrant having the value of θ to be 2 π − θ (VIII) For the given complex number, using (I), (III) to get, tan θ = y x tan θ = 2 2 tan θ = 1 For the given complex number, Using (V), θ = π 4 (IX) Using (II), (IV), and (IX), to get, z 1 = 2 2 ( cos π 4 + i sin π 4 ) The polar form of the complex number is z 1 = 2 2

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ISBN: 9780134469140

Author: Robert F. Blitzer

Publisher: PEARSON

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Chapter 6.5, Problem 51PE

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To calculate: The division of the complex numbers z1=2+2i,z2=1+i in polar form.

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Chapter 6.5, Problem 50PE

Chapter 6.5, Problem 52PE

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The division of the complex numbers z 1 = 2 + 2 i , z 2 = 1 + i in polar form.

Solution Summary: The author calculates the division of two complex numbers in the polar form, lz_1=2+2i.

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To calculate: The division of the complex numbers z1=2+2i,z2=1+i in polar form.

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Chapter 6 Solutions