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The complex number z = e j 2 π / N is illustrated in [link] . It lies on the unit circle at angle θ = 2 π / N . When this number is raised to the n t h power, the result is z n = e j 2 π n / N . This number is also illustrated in [link] . When one of the complex numbers e j 2 π n / N is raised to the N t h power, the result is

( e j 2 π n / N ) N = e j 2 π n = 1 .
Figure one is a graph of a circle on a cartesian graph with two line segments from the origin to points on the circle. Above the circle is an expression that reads, (e^j2πn/N)^N = e^(j2πn) = 1. The first line segment begins at the origin and terminates at a point on the circle in the second quadrant of the graph. The point is labeled e^(j2πn/N). The second line segment begins at the origin and terminates at a point on the circle in the first quadrant of the graph. The point is labeled e^(j2π/N). Figure one is a graph of a circle on a cartesian graph with two line segments from the origin to points on the circle. Above the circle is an expression that reads, (e^j2πn/N)^N = e^(j2πn) = 1. The first line segment begins at the origin and terminates at a point on the circle in the second quadrant of the graph. The point is labeled e^(j2πn/N). The second line segment begins at the origin and terminates at a point on the circle in the first quadrant of the graph. The point is labeled e^(j2π/N).
The Complex Numbers e j 2 π / N and e j 2 π n / N

We say that e j 2 π n / N is one of the N t h roots of unity, meaning that e j 2 π n / N is one of the values of z for which

z N - 1 = 0 .

There are N such roots, namely,

e j 2 π n / N , n = 0 , 1 , ... , N - 1 .

As illustrated in [link] , the 12 t h roots of unity are uniformly distributed around the unit circle at angles 2 π n / 12 . The sum of all of the N t h roots of unity is zero:

S N = n = 0 N - 1 e j 2 π n / N = 0 .

This property, which is obvious from [link] , is illustrated in [link] , where the partial sums S k = n = 0 k - 1 e j 2 π n / N are plotted for k = 1 , 2 , ... , N .

Figure two is a circle with 12 line segments from the origin to points on the circle at every incremental 30-degree mark. Starting from the point horizontal and to the right at the 0-degree mark, the labeled points along the curve at the end of the line segments read as follows (in a counter clockwise direction): e^(j2π 0/12), e^(j2π/12), e^(j2π 2/12), e^(j2π 3/12), e^(j2π 4/12), e^(j2π 5/12), e^(j2π 6/12), e^(j2π 7/12), e^(j2π 8/12), e^(j2π 9/12), e^(j2π 10/12), e^(j2π 11/12). Figure two is a circle with 12 line segments from the origin to points on the circle at every incremental 30-degree mark. Starting from the point horizontal and to the right at the 0-degree mark, the labeled points along the curve at the end of the line segments read as follows (in a counter clockwise direction): e^(j2π 0/12), e^(j2π/12), e^(j2π 2/12), e^(j2π 3/12), e^(j2π 4/12), e^(j2π 5/12), e^(j2π 6/12), e^(j2π 7/12), e^(j2π 8/12), e^(j2π 9/12), e^(j2π 10/12), e^(j2π 11/12).
Roots of Unity

These partial sums will become important to us in our study of phasors and light diffraction in "Phasors" and in our discussion of filters in "Filtering" .

Figure three is a cartesian graph with a dodecagon in the first and second quadrants. The dodecagon is approximately 3.75 units in height. The left vertex of the base of the dodecagon is at the origin, and the base sits along the positive side of the horizontal axis. The angles of the dodecagon are shown to be measured as 2π/12. An arrow from the origin to the fifth vertex of the polygon in the top-right corner, and it is labeled S_5. Figure three is a cartesian graph with a dodecagon in the first and second quadrants. The dodecagon is approximately 3.75 units in height. The left vertex of the base of the dodecagon is at the origin, and the base sits along the positive side of the horizontal axis. The angles of the dodecagon are shown to be measured as 2π/12. An arrow from the origin to the fifth vertex of the polygon in the top-right corner, and it is labeled S_5.
Partial Sums of the Roots of Unity

Geometric Sum Formula. It is natural to ask whether there is an analytical expression for the partial sums of roots of unity:

S k = n = 0 k - 1 e j 2 π n / N .

We can imbed this question in the more general question, is there an analytical solution for the “geometric sum”

S k = n = 0 k - 1 z n ?

The answer is yes, and here is how we find it. If z = 1 , the answer is S k = k . If z 1 , we can premultiply S k by z and proceed as follows:

z S k = n = 0 k - 1 z n + 1 = m = 1 k z m = m = 0 k - 1 z m + z k - 1 = S k + z k - 1 .

From this formula we solve for the geometric sum:

S k = 1 - z k 1 - z z 1 k , z = 1 .

This basic formula for the geometric sum S k is used throughout electromagnetic theory and system theory to solve problems in antenna design and spectrum analysis. Never forget it.

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Source:  OpenStax, A first course in electrical and computer engineering. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10685/1.2
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