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  "body": "\n## Statement\n\nLet $n \\ge 2$ be an integer and let $\\zeta$ be a [[primitive root of unity|0J817RTVG6]].\nThen\n$$\\sum_{k=0}^{n-1} \\zeta^k = 1 + \\zeta + \\zeta^2 + \\dots + \\zeta^{n-1} = 0.$$\n\n## Remarks\n\nProof: $\\zeta \\neq 1$, so the [[geometric sum formula|03X8V7D64M]] gives $\\frac{\\zeta^n - 1}{\\zeta - 1} = 0$.\nThus the sum of all the $n$th roots of unity is $0$ for $n \\ge 2$. For $n = 1$ the sum is $1$.\n\n## Explanations\n\n### L1 human\n\nThe $n$th roots of unity are the corners of a regular polygon with center $0$.\nBy symmetry, their center of mass is $0$, and so their sum is $0$.\nFor $n = 4$: $1 + i + (-1) + (-i) = 0$.\n",
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