Nth Root of Unity in Complex Numbers
We know that the general form of a complex number is x + iy.
Comparing the nth root of unity to a general complex number we get,
x + iy = [cos (2mπ/n) + i sin (2mπ/n)]
⇒ x = cos (2mπ/n)…(i)
and y = sin (2mπ/n)…(ii)
Squaring and adding (i) and (ii) we get,
x2 + y2 = cos2(2mπ/n) + sin2(2mπ/n) = 1
The above equation is the equation of the circle with the centre at origin (0,0) and radius 1.
If we represent the complex number as ω,
ω = e(i2mπ/n)
Taking power n both sides,
(ω)n = e(i2mπ/n)n
⇒ ωn = ei2mπ
This gives the nth root of unity taking n ≥ 0, we get the root of unity as,
1, ω, ω2, ω3,…ωn-1
These roots can be represented in a unit circle in a complex plane as,
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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.