# 4.5: Roots of Unity and Related Topics

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The complex number \(z=e^{j2π/N}\) is illustrated in Figure \(\PageIndex{1}\). It lies on the unit circle at angle \(θ=2π/N\). When this number is raised to the n^{th} power, the result is \(z^n=e^{j2πn/N}\). This number is also illustrated in Figure \(\PageIndex{1}\). When one of the complex numbers \(e^{j2πn/N}\) is raised to the N^{th} power, the result is

\[(e^{j2πn/N})^N=e^{j2πn}=1\]

We say that \(e^{j2πn/N} is one of the N^{th} roots of unity, meaning that \(e^{j2πn/N}\) is one of the values of z for which

\[z^N−1=0\]

There are N such roots, namely,

\[e^{j2πn/N},n=0,1,...,N−1\]

As illustrated in the Figure \(\PageIndex{2}\), the 12^{th} roots of unity are uniformly distributed around the unit circle at angles \(2πn/12\). The sum of all of the N^{th} roots of unity is zero:

\[S_N=\sum_{n=0}^{N−1}e^{j2πn/N}=0\]

This property, which is obvious from the Figure, is illustrated in Figure, where the partial sums \(S_k=\sum_{n=0}^{k−1} e^{j2πn/N}\) are plotted for \(k=1,2,...,N\).

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".

## Geometric Sum Formula

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

\[S_k=\sum_{n=0}^{k−1}e^{j2πn/N}\]

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

\[S_k=\sum_{n=0}^{k−1}z^n?\]

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:

\[\begin{align*} zS_k &=\sum^{k−1}_{n=0}z^{n+1}=\sum^k_{m=1}z^m \\[4pt] &=\sum^{k−1}_{m=0}z^m+z^k−1 \\[4pt] &=S_k+z^k−1 \end{align*}\]

From this formula we solve for the geometric sum:

\[S_k=\begin{matrix}\frac {1−z^k} {1−z} & z≠1\\k & z=1\end{matrix}\]

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

Exercise \(\PageIndex{1}\)

Find formulas for \(S_k=\sum_{n=0}^{k−1}e^{jnθ}\) and for \(S_k=\sum_{n=0}^{k−1}e^{j2π/Nn}\).

Exercise \(\PageIndex{2}\)

Prove \(\sum_{n=0}^{N−1}e^{j2πn/N}=0\).

Exercise \(\PageIndex{3}\)

Find formulas for the magnitude and phase of the partial sum \(S_k=\sum_{n=0}^{k−1}e^{j2πn/N}\).

Exercise \(\PageIndex{4}\)

(MATLAB) Write a MATLAB program to compute and plot the partial sum \(S_k=\sum_{n=0}^{k−1}e^{j2πn/N}\) for \(k=1,2,...,N\). You should observe the last Figure.

Exercise \(\PageIndex{5}\)

Find all roots of the equation \(z^3+z^2+3z−15=0\).

Exercise \(\PageIndex{6}\)

Find \(c\) so that \((1+j)\) is a root of the equation \(z^{17}+2z^{15}−c=0\).