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  "body": "\n## Proof\n\n1. Let $z \\neq 0$, let $r = |z|$ be its [[modulus|0KW2X0TFBX]] and let $\\theta = \\operatorname{Arg} z$.\n2. By the [[principal value of the argument|0SQSQP748R]], $\\theta$ is an [[argument|0ZZRSX91QR]] of $z$.\n3. By the definition of an argument, $z = r (\\cos \\theta + i \\sin \\theta)$.\n4. The number $\\theta$ is real. For real arguments, the power series of $\\cos$ and $\\sin$ are the real [[sine and cosine|0WB57D9488]].\nSo [[Euler's formula|04R3CY41VJ]] with $z = \\theta$ gives $e^{i \\theta} = \\cos \\theta + i \\sin \\theta$.\n5. Combining steps 3 and 4, $z = r (\\cos \\theta + i \\sin \\theta) = r e^{i \\theta}$.\n\n## Idea\n\nThe first equality is just the meaning of the word argument, once we know that the principal argument is an argument.\nThe second equality rewrites $\\cos \\theta + i \\sin \\theta$ with Euler's formula.\n",
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