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  "body": "\n## Proof\n\nWe use the [[construction of the complex numbers as pairs of reals|0CBAB9QQ9X]].\n\n1. In this construction, a [[complex number|09FTAET5Z4]] is an [[ordered pair|0JQWWFCS9R]] $(a, b)$ of real numbers.\nAddition is $(a, b) + (c, d) = (a + c, b + d)$, and multiplication is $(a, b)(c, d) = (a c - b d, a d + b c)$.\n2. A real number $a$ is the pair $(a, 0)$, and the [[imaginary unit|0ESCX36MMC]] is $i = (0, 1)$.\n3. Let $a$ and $b$ be real. Then $b i = (b, 0)(0, 1) = (b \\cdot 0 - 0 \\cdot 1, \\, b \\cdot 1 + 0 \\cdot 0) = (0, b)$.\nSo $a + b i = (a, 0) + (0, b) = (a, b)$.\n4. By step 3 and the definition of the [[real and imaginary parts|02ZE6JHJ7G]], the pair $z = (a, b)$ has $\\operatorname{Re} z = a$ and $\\operatorname{Im} z = b$.\nThese parts are well defined. If also $z = a' + b' i$ with $a', b'$ real, then $(a, b) = (a', b')$ by step 3, so $a = a'$ and $b = b'$.\n5. Let $z = (a, b)$ and $w = (c, d)$. By step 4, $\\operatorname{Re} z = a$, $\\operatorname{Im} z = b$, $\\operatorname{Re} w = c$ and $\\operatorname{Im} w = d$.\n6. By the defining property of ordered pairs, $(a, b) = (c, d)$ holds if and only if $a = c$ and $b = d$.\n7. By steps 5 and 6, $z = w$ if and only if $\\operatorname{Re} z = \\operatorname{Re} w$ and $\\operatorname{Im} z = \\operatorname{Im} w$.\n\n## Idea\n\nA complex number is nothing more than its pair of real coordinates $(a, b)$.\nTwo pairs are equal exactly when their first entries agree and their second entries agree.\n",
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