Question

What is the sum of the nth roots of unity? What is their product if n is odd? If n is even?

Short answer

Sum of nth root of unity =0

Product of nth root of unity, when number of terms are even =-1

Product of nth root of unity, when number of terms are odd =1

Step-by-step solution

Sum of  n th root of unity

$$\begin{gathered} \mathrm{The}n\mathrm{th}\mathrm{roots}\mathrm{of}\mathrm{unity}\mathrm{are}1,\mathrm{W},{\mathrm{W}}^2,{\mathrm{W}}^3,....{\mathrm{W}}^{\mathrm{N}-1} \\ \mathrm{Where}, \\ \mathrm{W}={\mathrm{e}}^{\frac{2\mathrm{\pi }}{\mathrm{n}}}={\cos }^{\frac{2\mathrm{\pi }}{\mathrm{n}}}\mathrm{\theta }+\mathrm{i}{\sin }^{\frac{2\mathrm{\pi }}{\mathrm{n}}}\mathrm{\theta } \end{gathered}$$

The sum of nth roots of unity is:

$1,\mathrm{W},{\mathrm{W}}^2,{\mathrm{W}}^3,.....{\mathrm{W}}^{\mathrm{N}-1}$ …… (1)

As per the geometric series, the sum of n terms of $1,\mathrm{W},{\mathrm{W}}^2,{\mathrm{W}}^3,.....{\mathrm{W}}^{\mathrm{N}-1}$is represented as .

So, equation (1) can be represented as follows:

Sum of n^th root of unity= $\frac{1-w^n}{1-w}$

$$\begin{gathered} =\frac{1-\left(\cos {\displaystyle \frac{2\pi }{n}}+i\sin \frac{2\pi }{n}\right)}{1-\left(\cos \frac{2\pi }{n}+i\sin \frac{2\pi }{n}\right)} \\ =\frac{1-\left(\cos {\displaystyle 2\pi }+i\sin 2\pi \right)}{1-\left(\cos \frac{2\pi }{n}+i\sin \frac{2\pi }{n}\right)} \\ =\frac{1-\left(1+0\right)}{1-\left(\cos \frac{2\pi }{n}+i\sin \frac{2\pi }{n}\right)} \\ =\frac{1-1}{1-\left(\cos \frac{2\pi }{n}+i\sin \frac{2\pi }{n}\right)} \\ =0 \end{gathered}$$

Product of nth root of unity 

The production of th root unity are following: $1,\mathrm{W},{\mathrm{W}}^2,{\mathrm{W}}^3,....{\mathrm{W}}^{\mathrm{N}-1}...\left(3\right)$... (3)

As per the arithmetic series, the product of n terms of $1,\mathrm{W},{\mathrm{W}}^2,{\mathrm{W}}^3,....{\mathrm{W}}^{\mathrm{N}-1}$is

represented as role="math" localid="1658921404734" ${\mathrm{W}}^{\frac{\mathrm{n}\left(\mathrm{n}-1\right)}{2}}$

So, calculation (3) is present as follows:

Product of th root of unity $={\mathrm{W}}^{\frac{\mathrm{n}\left(\mathrm{n}-1\right)}{2}}$

$={\left(\cos \frac{2\prod }{\mathrm{n}}+\mathrm{i}\sin \frac{2\prod }{\mathrm{n}}\right)}^{\frac{\mathrm{n}\left(\mathrm{n}-1\right)}{2}}$ ... (4)

If is even, then

Product of nth root of unity =-1 … (5)

If n is odd, then

Product of nth root of unity =1 … (6)

Moreover,

• This summation of such nth component of unity is given by Equation (2), is equal to 0.
• If "n" is even, this products of both the nth root for unity is obtained from Equation (5), is equal to -1.
• If "n" is odd, the composite of nth roots of unity may be calculated using Equation (6), is equal to 1.