nth Roots of a Complex Number and nth Roots of Unity

A positive real number $z_0$ has exactly one positive real $n^{th}$ root, but complex numbers behave differently: a non-zero complex number $z_0$ has exactly $n$ distinct complex numbers $\omega$ satisfying $\omega^n=z_0$, and all $n$ of them deserve to be called '$n^{th}$ roots' of $z_0$ (written $z_0^{1/n}$ or $\sqrt[n]{z_0}$). De Moivre's theorem for rational indices is exactly the tool that produces all of them at once.

Write $z_0$ in polar form as $z_0=r_0(\cos\theta_0+i\sin\theta_0)$ with $r_0=|z_0|>0$. For $k=0,1,2,\dots,(n-1)$ define

Each $a_k$ really does satisfy $a_k^n=z_0$: raising it to the $n^{th}$ power gives modulus $r_0$ and argument $\theta_0+2k\pi$, and since sine and cosine repeat every $2\pi$, that argument is equivalent to $\theta_0$. Conversely, every complex number whose $n^{th}$ power is $z_0$ turns out to coincide with one of these $a_k$ — matching moduli forces $|\omega|=r_0^{1/n}$, and matching arguments (up to a multiple of $2\pi$) forces the argument to be $(\theta_0+2k\pi)/n$ for some integer $k$, which by the division algorithm always reduces to one of $k=0,\dots,n-1$. And the $n$ values $a_0,\dots,a_{n-1}$ are genuinely distinct, because their arguments $(\theta_0+2k\pi)/n$ land at $n$ different points spread evenly around the circle. So $a_0,a_1,\dots,a_{n-1}$ are exactly the $n$ distinct $n^{th}$ roots of $z_0$ — no more, no fewer.

Geometrically this is a clean picture: all $n$ roots sit on the circle of radius $r_0^{1/n}$ centred at the origin, and consecutive roots are always $2\pi/n$ apart in angle. So the $n$ roots are the vertices of a regular $n$-gon inscribed in that circle.

nth roots of unity. The most important special case is $z_0=1$ (so $r_0=1,\ \theta_0=0$). Writing $\omega = \cos\dfrac{2\pi}{n}+i\sin\dfrac{2\pi}{n}$, the formula above collapses to $a_k=\omega^k$, so the $n^{th}$ roots of unity are precisely

They are $n$ equally spaced points on the unit circle, forming a regular $n$-gon with one vertex fixed at $1$ (i.e. at angle $0$) — this is the geometric picture behind, for example, the 8 vertices of a regular octagon when $n=8$. A few facts about this list are used constantly:

• They form a geometric progression with common ratio $\omega$, since each is $\omega$ times the previous one.
• Their sum is always $0$ (for $n>1$): using the GP sum formula, $1+\omega+\cdots+\omega^{n-1} = \dfrac{1-\omega^n}{1-\omega} = \dfrac{1-1}{1-\omega}=0$, since $\omega^n=1$.
• Their product is $(-1)^{n-1}$: the product is $\omega^{1+2+\cdots+(n-1)} = \omega^{n(n-1)/2} = cis\big((n-1)\pi\big) = (-1)^{n-1}$.

Worked example. Find all values of $(1+i)^{1/4}$ and sketch what they look like geometrically. …