How to Find nth Root of Unity?
Nth root of unity can be easily found by finding the solution to the equation,
zn = 1
In polar form, we write this equation as,
zn = cos 0 + i sin 0
In general form,
zn = cos (0+2mπ) + i sin (0+2mπ) (where m∈N)
Taking nth root on both sides,
z = [cos (2mπ) + i sin (2mπ)]1/n
Using DeMoivre’s Theorem
z = [cos (2mπ/n) + i sin (2mπ/n)]
This can be represented in Euler Form,
z = e(i2mπ/n)
This is the nth root of unity for m ∈ N
Nth Root
Nth root of unity is the root of unity when taken which on taking to the power n gives the value 1. Nth root of any number is defined as the number that takes to the power of n results in the original number. For example, if we take the nth root of any number, say b, the result is a, and then a is raised to n power and will get b.
We define the nth root of any number as suppose we take nth power of any number ‘a’
an = b
then, the nth root of ‘b’ is ‘a’ we represent this as,
n√b = a
We can also check for the nth root of unity as,
zn = 1
then, the nth root of ‘1‘ is ‘z‘ we represent this as,
n√1 = z
In this article, we will learn about, the nth root of any number, the nth root of unity, and others in detail.