The $n$th roots of unity are the unique solution to the equation $\zeta^{n} = 1$. In the [[Real Numbers]], the solution is obviously $1$ (and $-1$ when $n$ is even), but the roots of unity includes both these and members in the [[Complex Plane]]. These numbers all lie on the [[Unit Circle]], and can be derived by this formula using the [[Complex Exponential|complex exponential]]: $\huge \begin{align} 1 &= \zeta^{n} \\ \zeta &= \set{ 1, e^{ {2\pi i}/{n} }, e^{ {(2)2\pi i}/{n} }, e^{ {(3)2\pi i}/{n} }, \cdots, e^{ {(n-1)2\pi i}/{n} } } \\ \zeta &= \setbuild{k \in [0, n)}{e^{k2\pi i/n}} \end{align} $ Typically the value of $\zeta$ itself is not important, but the key property we care about is that successive powers of $\zeta$ will rotate until it returns back to $\zeta$. For example, if $\zeta$ is a $n$th root of unity, then $\zeta^{n}=1$, therefore $\zeta^{n+1}=\zeta$. ## Modular Arithmetic Connectios The roots of unity play an important role in many branches of math due to their ability to describe [[Frequency]] problems. An example of this is showing that the [[Group]] structure of a [[Modular Ring]] of [[Modulo]] $m$ is [[Isomorphism|Isomorphic]] to the [[Multiplicative Group]] of the [[Roots of Unity]]. This means that our modulo ring operations on the following relation: $\huge \begin{align} a &\equiv b \pmod{m} \\ a-b &\mid m \\ a \op{\,mod\,} m &\equiv b \op{\,mod\,} m \end{align} $ This means this operation turns all [[Natural Numbers]] into the following [[Set|set]]: $\huge M = \set{ 0, 1,\dots,m-1} $ We can represent how addition works between each element on this set directly, but we can draw an analogue ([[Isomorphism]]) using the roots of unity. For $m=3$, we can translate this as the following: $\huge \begin{align} M &= \set{0,1,2} \\ Z &= \left\{e^{ \frac{0}{3} 2\pi i }, e^{\frac{1}{3}2\pi i}, e^{\frac{2}{3}2\pi i} \right\} \\ &= \left\{ \zeta^{0}, \zeta^{1}, \zeta^{2}\right\} \end{align} $ This converts the group of integers $0,1,2$ under addition [[Modulo]] 3 into a new [[Group]] of $\set{\zeta^{0},\zeta^{1},\zeta^{2}}$ under multiplication of $\mathbb{C}$. During algabraic manipulations for the second [[Set|set]] can be easier and more powerful in many situations (see [[Generating Function]]), which shows is usages in many fields such as [[Number Theory]].