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  "body": "\n## Statement\n\nFor all complex numbers $z$ and $w$,\n$$|z w| = |z| \\, |w|,$$\nwhere $|\\cdot|$ is the [[modulus|0KW2X0TFBX]].\n\n## Remarks\n\nProof: $|z w|^2 = z w \\overline{z w} = (z \\bar{z})(w \\bar{w}) = |z|^2 |w|^2$, and both sides are nonnegative.\nFor $z = a + b i$ and $w = c + d i$, this is the identity $(a^2 + b^2)(c^2 + d^2) = (a c - b d)^2 + (a d + b c)^2$.\nOn the real numbers, it gives that the [[absolute value is multiplicative|0GMV7GP8W0]].\n\n## Explanations\n\n### L1 human\n\nThe length of a product is the product of the lengths.\nFor example, $|(1 + i)(1 - i)| = |2| = 2$, and $|1 + i| \\cdot |1 - i| = \\sqrt{2} \\cdot \\sqrt{2} = 2$.\n",
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