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  "body": "\n## Proof\n\n1. Let $x$ and $y$ be real numbers, and let $z = x + y i$.\n2. Since $x$ and $y$ are real, the definition of the [[real and imaginary parts|02ZE6JHJ7G]] gives $\\operatorname{Re} z = x$ and $\\operatorname{Im} z = y$.\n3. By the definition of the [[modulus|0KW2X0TFBX]], $|z| = \\sqrt{(\\operatorname{Re} z)^2 + (\\operatorname{Im} z)^2}$.\n4. Substituting step 2 into step 3 gives $|x + y i| = \\sqrt{x^2 + y^2}$.\n\n## Idea\n\nThe modulus is defined through the real and imaginary parts.\nWhen $x$ and $y$ are real, these parts of $x + y i$ are simply $x$ and $y$.\n",
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