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  "body": "\n## Statement\n\nLet $z$ be a complex number with $z \\neq 0$.\nThen\n$$z^{-1} = \\frac{\\bar{z}}{|z|^2}.$$\nWith $z = a + b i$ and $a, b$ real, this is $z^{-1} = \\dfrac{a}{a^2 + b^2} - \\dfrac{b}{a^2 + b^2} \\, i$.\n\n## Remarks\n\nProof: $z \\bar{z} = |z|^2$ by the formula for [[a complex number times its conjugate|0ZB7J071WQ]], and $|z|^2 \u003e 0$ because $z \\neq 0$. Divide by $|z|^2$.\nTo compute a quotient $w / z$, multiply the numerator and the denominator by $\\bar{z}$.\n\n## Explanations\n\n### L1 human\n\nTo divide by a complex number, multiply above and below by its conjugate. Then the denominator becomes a positive real number.\nFor example, $\\dfrac{1}{1 + i} = \\dfrac{1 - i}{(1 + i)(1 - i)} = \\dfrac{1 - i}{2}$.\n",
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