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  "body": "\n## Statement\n\nLet $x$ and $y$ be real numbers.\nThen the [[modulus|0KW2X0TFBX]] of the complex number $x + y i$ is\n$$|x + y i| = \\sqrt{x^2 + y^2}.$$\n\n## Remarks\n\nThis follows from the definition of the modulus, because $\\operatorname{Re}(x + y i) = x$ and $\\operatorname{Im}(x + y i) = y$ for real $x$ and $y$.\nThe hypothesis that $x$ and $y$ are real is necessary. For example, $|i + i \\cdot i| = |-1 + i| = \\sqrt{2}$, but $\\sqrt{i^2 + i^2} = \\sqrt{-2}$ is not even real.\n\n## Explanations\n\n### L1 human\n\nTo find the length of a complex number, square its two real coordinates, add them, and take the square root.\nFor $1 - i$, this gives $\\sqrt{1 + 1} = \\sqrt{2}$.\n",
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