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  "body": "\n## Statement\n\nLet $z$ be a complex number, let $\\bar{z}$ be its [[conjugate|0BJV24C4SY]] and let $|z|$ be its [[modulus|0KW2X0TFBX]].\nThen\n$$z \\bar{z} = |z|^2.$$\nIn particular, $z \\bar{z}$ is a nonnegative real number.\n\n## Remarks\n\nWith $z = a + b i$: $(a + b i)(a - b i) = a^2 - (b i)^2 = a^2 + b^2$.\nThis identity gives the formula for the [[inverse of a complex number|0FQRAV5SKH]].\nIt also gives a quick proof that the [[modulus is multiplicative|0HYNZE940T]].\n\n## Explanations\n\n### L1 human\n\nWhen you multiply a complex number by its conjugate, the imaginary parts cancel.\nWhat remains is the square of the length of the number, for example $(3 + 4 i)(3 - 4 i) = 9 + 16 = 25$.\n",
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