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  "body": "\n## Statement\n\nLet $z$ and $w$ be complex numbers with $z \\neq 0$ and $w \\neq 0$.\nIf $\\theta$ is an [[argument|0ZZRSX91QR]] of $z$ and $\\varphi$ is an argument of $w$, then $\\theta + \\varphi$ is an argument of $z w$.\nThus $\\arg(z w) \\equiv \\arg z + \\arg w \\pmod{2 \\pi}$.\n\n## Remarks\n\nProof: write $z = |z| e^{i \\theta}$ and $w = |w| e^{i \\varphi}$. Then $z w = |z| |w| e^{i (\\theta + \\varphi)}$ by the [[product of exponentials|0WM7DJCM6G]].\nFor the principal values, the equality can fail. For example, $\\operatorname{Arg}(-1) + \\operatorname{Arg}(-1) = 2\\pi$, but $\\operatorname{Arg}((-1)(-1)) = 0$.\n\n## Explanations\n\n### L1 human\n\nWhen you multiply two complex numbers, the angles add.\nFor example, $i$ points at $90$ degrees, and $i \\cdot i = -1$ points at $180$ degrees.\n",
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