nth Root of Unity

In mathematics, particularly in complex number theory, the concept of the nth root of unity is fundamental. An nth root of unity is a complex number that, when raised to the power of n, equals 1. In other words, if $\zeta$ is an nth root of unity, then $\zeta^n = 1$. These roots of unity have important applications in various fields such as number theory, algebra, and the discrete Fourier transform in signal processing.

Definition

The nth roots of unity are the solutions to the equation:

$$ x^n = 1 $$

where n is a positive integer. The solutions to this equation are complex numbers and are evenly distributed on the unit circle in the complex plane.

Formula

The general formula for the nth roots of unity is given by:

$$ \zeta_k = \cos\left(\frac{2k\pi}{n}\right) + i\sin\left(\frac{2k\pi}{n}\right) $$

where $i$ is the imaginary unit, $k$ is an integer ranging from 0 to $n-1$, and $\zeta_k$ is the kth nth root of unity.

Properties

The nth roots of unity have several important properties:

1. Symmetry: They are symmetrically distributed on the unit circle in the complex plane.
2. Multiplicative Closure: The product of any two nth roots of unity is also an nth root of unity.
3. Conjugates: The complex conjugate of an nth root of unity is also an nth root of unity.
4. Sum: The sum of all nth roots of unity is zero, except when $n=1$.

Table of Differences and Important Points

Property	Description
Distribution	The nth roots of unity are evenly spaced on the unit circle.
Principal Root	The principal nth root of unity is $\zeta_0 = 1$.
Powers	Raising an nth root of unity to the power of n always yields 1.
Summation	The sum of all nth roots of unity is 0 if $n > 1$.

Examples

Example 1: 4th Roots of Unity

Find the 4th roots of unity.

Solution:

Using the formula, we calculate the roots for $k = 0, 1, 2, 3$:

$$ \zeta_0 = \cos(0) + i\sin(0) = 1 $$ $$ \zeta_1 = \cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right) = i $$ $$ \zeta_2 = \cos(\pi) + i\sin(\pi) = -1 $$ $$ \zeta_3 = \cos\left(\frac{3\pi}{2}\right) + i\sin\left(\frac{3\pi}{2}\right) = -i $$

The 4th roots of unity are $1, i, -1, -i$.

Example 2: Sum of 3rd Roots of Unity

Calculate the sum of all 3rd roots of unity.

Solution:

The 3rd roots of unity are given by $k = 0, 1, 2$:

$$ \zeta_0 = \cos(0) + i\sin(0) = 1 $$ $$ \zeta_1 = \cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right) = -\frac{1}{2} + i\frac{\sqrt{3}}{2} $$ $$ \zeta_2 = \cos\left(\frac{4\pi}{3}\right) + i\sin\left(\frac{4\pi}{3}\right) = -\frac{1}{2} - i\frac{\sqrt{3}}{2} $$

The sum is:

$$ \zeta_0 + \zeta_1 + \zeta_2 = 1 + \left(-\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) + \left(-\frac{1}{2} - i\frac{\sqrt{3}}{2}\right) = 0 $$

The sum of all 3rd roots of unity is 0.

Conclusion

The nth roots of unity are a set of complex numbers that are solutions to the equation $x^n = 1$. They have a symmetrical distribution on the unit circle and possess several interesting properties that make them useful in various mathematical and engineering applications. Understanding these roots is essential for anyone studying complex numbers or related fields.