We have reached the final lesson on complex numbers. As a continuation of the previous lesson, we will now investigate the nth roots on the complex plane, a journey of cosine functions and geometry. Hang on for the ride!
NOTE: This is a 2-part video lesson where the first part is in the lesson 'nth roots of a Complex Number'. I really hope you do watch both of them for a complete picture of the principle.
GEOMETRIC INTERPRETATION OF NTH ROOTS
May want to check out:
NTH ROOTS OF A COMPLEX NUMBER
Similar to:
1990 AHSME #22 sol1
VIDEO LECTURE
MAIN CONCEPTS
Magnitude of nth roots is  and angle between roots is  .Get the Printer Friendly Version COMMENTS
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LESSON
In the previous lesson, we establish that
for each We will now consider the geometric interpretation of such a principle. Let us assume that the given complex number has magnitude of 4 and an argument of
Now let’s say we want to find the 6th root of
and by finding the first 6th root of
At this juncture, we want to suspect where the next 6th root would be. We shall first note that the magnitude would be
which we represent as
In fact which is independent of
And this is the reason why the values of k are from 0 to (n-1) because any further values we take will simply be repeating the cycle of the nth roots starting back with We conclude by saying the All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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.
or written in polar form,
. So using the principle, we have
we have
and an argument of 
. We shall represent this first 6th root on the complex plane. 
simply because the value of 
remains the same regardless the value of k. Now from the formula, we will calculate the angle between the first and second 6th root.
, the argument of the given complex number. We see from here that, regardless of the complex number 
given, the nth roots will be an angle of 
apart. For the case of finding the 6th roots, this angle is 
. And keeping in mind that they will have the same magnitude 
, we can now draw the 6th roots on the complex plane. 
.
th roots of a complex number will lie on a circle with radius 
