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  "body": "\n## Statement\n\nLet $n$ be a positive integer.\nA complex number $z$ is an $n$th [[root of unity|0S61N75H45]] if and only if\n$$z = e^{2 \\pi i k / n} = \\cos \\frac{2 \\pi k}{n} + i \\sin \\frac{2 \\pi k}{n}$$\nfor some integer $k$ with $0 \\le k \u003c n$.\n\n## Remarks\n\nProof: by [[de Moivre's formula|03MTBVG3VE]], each such $z$ satisfies $z^n = e^{2 \\pi i k} = 1$. Conversely, $z^n = 1$ forces $|z| = 1$ and $n \\arg z \\in 2 \\pi \\mathbb{Z}$.\nFor $n = 3$, the roots are $1$ and $-\\frac{1}{2} \\pm \\frac{\\sqrt{3}}{2} i$.\n\n## Explanations\n\n### L1 human\n\nThe $n$th roots of unity sit on the unit circle at equal angles of $2\\pi/n$, starting at $1$.\nFor $n = 6$, they are the six corners of a regular hexagon.\n",
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