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Roots of Unity

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Complex Numbers

N-th root of unity

The n-th roots of unity are the complex solution to the equation: xn = 1. This equation asks for all complex numbers z such that raising z to the power n yields 1. This can be approached using Euler’s formula, which connects complex exponentials with trigonometric functions.

In the real number system, the only solution to this equation is x = 1, since 1 is the only real number that, when raised to any power, results in 1.

However, in the complex domain, there are n distinct solutions to this equation, distributed symmetrically on the unit circle (the set of points in the complex plane at a distance of 1 from the origin). These roots are known as the n-th roots of unity.

Finding the N-th Roots of Unity

The number 1 can be represented using Euler’s formula by choosing θ = 0: 1 = $e^{i·0} = cos(0)+ isin(0) = 1 + i·0$. The complex equation zn = 1 can be rewritten using exponential form: $z^n = e^{i·0} = e^{i·2πk}$ for any integer k because adding multiples of 2π to the angle 0 does not change the value of e due to the periodicity of sine and cosine functions.

$z^n = e^{i·2πk}$ ⇒[Solving for z:] $z = e^{\frac{i·2πk}{n}}$ for k = 0, 1, 2, ···, n -1. There are n distinct solutions corresponding to k = 0 through k = n-1. Using Euler’s formula, each root can be expressed as: $z_k = cos(\frac{i·2πk}{n})+isin(\frac{i·2πk}{n})$ where k = 0, 1, 2, ···, n -1.

Conclusion: The n-th roots of unity are the solution to the equation zn = 1, where z is a complex number. These roots are given by the formula: $z_k = e^{\frac{2πik}{n}} = cos(\frac{2πk}{n}) + isin(\frac{2πk}{n})$ where k = 0, 1, 2, ···, n-1 and:

Examples of N-th Roots of Unity

k = 0, $z_0 = e^{\frac{2πi·0}{2}}=e^0=1$. k = 1, $z_1 = w_2 = e^{\frac{2πi·1}{2}}=e^{πi} = cos(π)+isin(π) = -1 + i·0 = -1.$

The square roots of unity are 1 and −1, lying on opposite ends of the real axis on the unit circle.

n = 3, k = 0, $z_0 = e^{\frac{2πi·0}{3}}=e^0=1, z_1 = w_3 = e^{\frac{2πi}{3}} = cos(\frac{2πi}{3}i)+isin(\frac{2πi}{3}i), w_3² = e^{\frac{4πi}{3}}$

k = 0, $z_0 = e^{\frac{2πi·0}{4}} = e^0 = 1$. k = 1, $z_1 = w_3 = e^{\frac{2πi·1}{4}} = e^{\frac{πi}{2}} = cos(\frac{π}{2})+isin(\frac{π}{2}) = 0 + i·1 = i$. k = 2, $z_2 = w_3² = e^{\frac{2πi·2}{4}}=e^{πi} = cos(π)+isin(π) = -1 + i·0 = -1$. k = 3, $z_3 = w_3³ = e^{\frac{2πi·3}{4}} = e^{\frac{3πi}{2}} = cos(\frac{3π}{2})+isin(\frac{3π}{2}) = 0 + i·(-1) = -i$

The fourth roots of unity are 1, i, −1, and -i, which are positioned at equal angles of π2 radians around the unit circle.

In the complex plane, the n-th roots of unity are arranged as the vertices of a regular n-sided polygon inscribed in the unit circle. Each vertex corresponds to one of the roots, and the angle between any two consecutive roots is $\frac{2π}{n}$ radians (Refer to Figure iv for a visual representation and aid in understanding it).

Properties of N-th Roots of Unity:

  1. Distinct Roots: All n roots z0, z1, ··, zn-1 are distinct.
  2. Equally Spaced on the Unit Circle: Each root is obtained by rotating the previous one by $\frac{2π}{n}$ radians, meaning they are evenly distributed around the circle.
  3. Product of Roots. The product of any two n-th roots of unity is also an n-th root of unity: $z_k·z_m = e^{\frac{2πik}{n}}e^{\frac{2πim}{n}} = e^{\frac{2πi(k+m)}{n}} = z_{(k + m)~ mod~ n}$
  4. Inverse of Each Root. Each n-th root of unity has an inverse, which is its complex conjugate: $z_k^{-1} = \bar z_k = e^{-\frac{2πik}{n}} = z_{n-k}$.
  5. Sum of All N-th Roots of Unity. The sum of all n-th roots of unity is zero: $\sum_{k=0}^{n-1} z_k = 0$.

    This property holds because the roots are symmetrically distributed around the origin in the complex plane, so their vector sum cancels out.

In modular arithmetic, the integers coprime (relatively prime) to n from the set { 0 , 1 , … , n − 1 } of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. This group is usually denoted as $\mathbb{Z}/n\mathbb{Z}$.

The set of complex numbers {$1,ζ_n,ζ_n^2, \ldots ,ζ_n^{n−1 }$} where $ζ_n = w_n = e^{\frac{2\pi i}{n} }$ and $\zeta_n^n = 1$ is cyclic, with $ζ_n$ as a generator.

There is an isomorphism between $\mathbb{Z}/n\mathbb{Z}$ and the group of nth roots of unity. Let $ϕ:(Z/nZ)^*→{ζ_n^k∣k∈Z}$ be defined by ϕ(a) = $\zeta_n^a$, e.g., $\phi(ab) = \zeta_n^{ab} = \zeta_n^a \cdot \zeta_n^b = \phi(a) \cdot \phi(b)$ (homomorphism)

Prime nth roots of z

Given a complex number w = se, β = Arg(w), we are looking at the solution set of the equation zn = w, where z = re.

zn = (re)n = rn(e)n = [De Moivre’s Theorem] rneinθ

Our goal is to identify all possible z that satisfies: se = rneinθ ⟷ s = rn, e = einθ ⟷[s = |w|, hence s >= 0] $r = \sqrt{s}, nθ-β = 2πk ⇒ θ = \frac{β + 2πk}{n}$ for some k ∈ ℤ

If w = zn, z ∈ $\sqrt[\frac{1}{n}]{w}$ (n roots of w)

We have already study that for this equation to hole, z = $\sqrt[n]{s}e^{i(\frac{β + 2πk}{n})}$ for some k ∈ ℤ

Next, considering that $e^{i(\frac{β + 2πk}{n})} = e^{i(\frac{β}{in})}e^{i(\frac{2πk}{n})}$ = [De Moivre’s Theorem] $e^{i(\frac{β}{n})}(e^{i(\frac{2π}{n})})^k$ = [Notation, the principal root of unity, $w_n=ζ_n=e^{\frac{2πi}{n}}$] $e^{i(\frac{Arg(w)}{n})}w_n^k$

z = $\sqrt[n]{s}e^{i(\frac{β + 2πk}{n})} = \sqrt[n]{|w|}e^{i(\frac{Arg(w)}{n})}w_n^k$ where 0 ≤ k ≤ n-1

Definition. The principal nth root of w can be given by: $\sqrt[n]{w} = \sqrt[n]{|w|}e^{i(\frac{Arg(w)}{n})}$ where $w^{\frac{1}{n}} = ${$\sqrt[n]{w}, w_n\sqrt[n]{w}, w_n²\sqrt[n]{w}, ..., w_n^{n-1}\sqrt[n]{w}$} is the set of nth roots of a complex number w = $|w| e^{i \text{Arg}(w)}$, where |w| is the magnitude (modulus) and $\text{Arg}(w)$ is the argument (angle) of w.

These roots are obtained by adding multiples of $\frac{2\pi}{n}$ to the argument of the principal root. Each root corresponds to rotating the principal nth root by $\frac{2\pi}{n}$ radians around the origin.

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