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  "body": "\n## Proof\n\n1. Let $z = a + b i$ with $a$ and $b$ real.\n2. By the definition of the [[complex conjugate|0BJV24C4SY]], $\\bar{z} = a + (-b) i$.\nSo $\\operatorname{Re} \\bar{z} = a$ and $\\operatorname{Im} \\bar{z} = -b$, by the definition of the [[real and imaginary parts|02ZE6JHJ7G]].\n3. By the definition of the [[modulus|0KW2X0TFBX]], $|\\bar{z}| = \\sqrt{a^2 + (-b)^2}$.\n4. Since $(-b)^2 = b^2$, this is $\\sqrt{a^2 + b^2}$, which is $|z|$ by the same definition.\nSo $|\\bar{z}| = |z|$.\n\n## Idea\n\nConjugation changes the sign of the imaginary part only.\nThe modulus depends on the square of that part, so the sign does not matter.\n",
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