SOLUTION: How does one determine the number of roots of unity? The solution for the following problems must be made using the radian not polar rectangular approach:
(1-i)^1/4
(1+i*sqr
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Question 211188This question is from textbook Visual Complex Analysis
: How does one determine the number of roots of unity? The solution for the following problems must be made using the radian not polar rectangular approach:
(1-i)^1/4
(1+i*sqrt(3)^7/2
(-i)^1/2
These problems I have to work are not from my text, but rather from the instructor. My textbook is:
Tristan Needham's - Visual Complex Analysis ISBN 978 0 19 853446 4 (pbk)
This question is from textbook Visual Complex Analysis
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
determine the number of roots of unity? The solution for the following problems must be made using the radian not polar rectangular approach:
(1-i)^1/4
------
On a coordinate system, plot the point (1,-1)
Notice that it is in the 4th quadrant.
--------------------------
r = sqrt(1 + 1) = sqrt(2)
theta = arctan(-1/1) = 315 degrees in the 4th quadrant.
---
(1-i)^(1/4) = r^(1/4)(cis[(315+(n(360)]/4) where n = 0,1,2,3
= 2^(1/8)cis(78.75 degrees) when n=0
= 2^(1/8)cis(78.75+90) when n = 1
= 2^(1/8)cis(78.75+180) when n = 2
= 2^(1/8)cis(78.75+270) when n = 3
=========================================
(1+i*sqrt(3)^7/2
Plot the point; it is in the 1st quadrant.
r = sqrt(1+3) = 2
theta = arctan(sqrt(3)) = 60 degrees
---
(1+i*sqrt(3))^(7/2) = 2^(7/2)cis[(60 + n*360)]/(7/2) when n = 0,1. etc.
= 2^(7/2)cis(17.14 degrees) when n = 0
= 2^(7/2)cis(17.14+102.86) when n = 1
etc.
============================
(-i)^1/2
Plot the point (0,-1)
r = sqrt(0+1) = 1
theta = arctan(-1/0) which is undefined but you can see that theta = 270
---------------
(-i)^(1/2)= 1cis(270+360n)/2
= cis(135) when n=0
= cis(135+180) when n = 1
====================================
Cheers,
Stan H.
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