nth root of unity

In the English language, the word unity has a variety of connotations, but it is likely best recognised for its most basic and plain definition, which is “the state of being one; oneness.” While the word has its own unique meaning in mathematics, the unique use does not travel too far from this definition, at least metaphorically. In actuality, the number “one” (1), the integer between zero (0) and two (2), is simply referred to as “unity” in mathematics.

nth root of unity: –

 The root of unity is usually a complex number that is raised to the power n (integer) and equals one. The Moivre number is another widely used name for the unit’s root. If n is a positive integer and Z is a number equal to the nth root of unity (1), then

        Zn = 1

Or, 11n= Z

As a result, the nth root of unity in the above equation equals Z. GP contains all of the n roots of the nth roots of unity (Geometric progression). When all the n roots of the nth root are added together, the result is zero. And if we add all the n roots of the nth roots together, unity equals (-1)n-1.

How can I find the nth Unity Root?

As previously stated, if ‘Z’ is the nth root of unity, it will meet the following conditions:

Zn = 1

As a result, the nth root of unity is the positive value of the integer.

We can now put the equation above in polar form:

Zn = cos 0 + i sin 0

Zn = cos (0+2kπ) + i (0+sin 2kπ) [where, k is an integer]

Taking the nth root on both sides now yields;

Z = (cos 2kπ + i sin 2kπ)1/n

We can find the nth root of unity using de Moivre’s theorem.

Z = (cos (2kπ/n) + i sin (2kπ/n)) = e(i2kπ/n) ; where k = 0 , 1, 2 , 3 , 4 , ……… , (n-1)

Only if Zn = 1 does the preceding equation reflect the nth root of unity.

As a result, each unity root becomes:

Z = cos [(2kπ)/n] + i sin[(2kπ)/n] ; where 0 ≤ k ≤ n-1

nth root of unity:-

A complex number’s generic form is given by:

x+iy

Where ‘x’ is the real part and ‘iy’ is the imaginary part.

The argand plane or the complex plane are commonly used to plot these complex values. When the roots of the unity equation are compared to the complex number form, we get:

x + iy = cos [(2kπ)/n] + I sin[(2kπ)/n]

x = cos [(2kπ)/n]

y = sin[(2kπ)/n]

So,

X2 + y2 = cos2[(2kπ)/n] + sin2[(2kπ)/n] = 1

As a result, it fulfils the circle-origin equation (0,0).

If Ω is used to represent a complex number, then;

Ω = e2πi/n = cos (2πi/n) + I sin (2πi/n)

Ωn = (e2πi/n)n = e2πi = 1

As a result, the nth root of unity is Ω.

The complex integers 1, 2,…., n-1 are nth roots of unity, according to deMoivre’s theorem. As a result, all complex numbers 1, 2,…., n-1 are points in a plane and vertices of a regular n-sided polygon inscribed in a unit circle.

Properties of nth root of unity:-

The nth root of unity has the following properties.

• The n roots of the nth roots unity are found on the perimeter of a circle with a radius of 1 and the origin as its centre (0,0).
• 1, -1/2+i√(3)/2, -1/2 – i√(3)/2 are the three cube roots of unity.
• When we multiply two imaginary cube roots, we get 1 as a result.
• The square of another is one of the imaginary cube roots of unity.
• All of unity’s nth roots add up to zero.

1 + [(-1 + √3i ) /2] + [(-1 – √3i ) /2] = 0

• The nth roots of unity are in geometric progression with a common ratio and are 1,ω,ω2 ,… …,ωn-1. As a result, 1+ ω + ω2 +… + ωn-1 = (1-ωn)/1- ω = 0 because n = 1 and 1 is a prime number.
• 1.ω.ω² … …ωn-1 = (-1)n-1 is the sum of all the nth roots of unity.

Conclusion:-

In general, the term root of unity, also known as a de Moivre number, refers to a complex number that, when multiplied by some integer n, equals 1.

If n is a positive integer, then ‘x’ is said to be an nth root of unity if the equation xn = 1 is satisfied. As a result, this equation has n roots, commonly known as the nth roots of unity.