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  "body": "\n## Statement\n\nLet $z$ be a complex number with $z \\neq 0$.\nThen $z$ has exactly one [[argument|0ZZRSX91QR]] $\\theta$ with $-\\pi \u003c \\theta \\le \\pi$.\nThis argument is the principal value of the argument, written $\\operatorname{Arg} z$.\n\n## Remarks\n\nProof: the arguments of $z$ form a set $\\theta_0 + 2 \\pi \\mathbb{Z}$, and each interval $(-\\pi, \\pi]$ of length $2\\pi$ contains exactly one element of such a set.\nFor example, $\\operatorname{Arg}(-1) = \\pi$, $\\operatorname{Arg}(-i) = -\\pi/2$ and $\\operatorname{Arg} x = 0$ for real $x \u003e 0$.\nSome authors use the interval $[0, 2\\pi)$ instead.\n\n## Explanations\n\n### L1 human\n\nMany angles describe the same direction, because full turns do not change it.\nThe principal value picks one of them, the one between $-180$ and $180$ degrees, with $180$ allowed and $-180$ not.\n",
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