Question

What value in the nth root of unity for any value n?.

Answer 1

Answer with explanation:

The equation for nth root of unity is

$Z^n=1\\\\Z=(1)^{\frac{1}{n}}\\\\Z=[cos (2k\pi+ 0)+i sin(2k\pi+ 0)]^{\frac{1}{n}}\\\\Z=cos(\frac{2k\pi}{n})+i sin(\frac{2k\pi}{n})=e^{\frac{2k\pi}{n}}$

Where,k=0,1,2,3......(n-1).

there will be n-roots of the above equation.

For,different values of k,there will be different Z.

$Z=1,e^{\frac{2\pi}{n}} ,e^{\frac{4\pi}{n}},e^{\frac{6\pi}{n}},e^{\frac{8\pi}{n}},....e^{\frac{2(n-1)\pi}{n}$

Used,Euler's Identity

$e^{i\alpha}=cos\alpha + isin\alpha$

Answer 2

For different values of k , the different roots of unity will be

Z = $\rm 1 , e^{2\pi / n} , \rm e^{4\pi / n}$....................

What is Euler's Identity ?

According to Euler Identity , $\rm e^{i\pi}$ = -1

It is asked to determine nth root of Unity or any value of n

Zⁿ = 1

Z = $\rm 1^{1/n}$

By Euler's Formula

Z = {  cos ( 2kπ +0 ) + isin ( 2kπ +0 )}

Z = cos ( 2kπ / n) + isin ( 2kπ / n)

From euler' identity this can be written as

Z = $\rm e^{2k\pi / n}$

where k = 0 , 1 , 2 , 3..... (n-1)

So , the roots of unity will be n different roots.

Therefore , For different values of k , the different roots are as follows

Z = $\rm 1 , e^{2\pi / n} , \rm e^{4\pi / n}$....................

To know more about Euler's Identity

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