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  "body": "\n## Summary\n\nThe complex numbers can be constructed as ordered pairs of [[real numbers|08J2M1JYAG]].\nOn $\\mathbb{R}^2$, define addition by $(a, b) + (c, d) = (a + c, b + d)$ and multiplication by $(a, b) \\cdot (c, d) = (a c - b d, a d + b c)$.\nWith these operations, $\\mathbb{R}^2$ is a field. It is the field $\\mathbb{C}$ of [[complex numbers|09FTAET5Z4]].\nThe pair $(a, 0)$ is the real number $a$, and the pair $(0, 1)$ is the [[imaginary unit|0ESCX36MMC]] $i$.\nThen $(a, b) = a + b i$, and the map $a + b i \\mapsto (a, b)$ is a bijection $\\mathbb{C} \\to \\mathbb{R}^2$.\n\n## Remarks\n\nThis construction shows that the complex numbers exist, without the assumption of a mysterious square root of $-1$.\nThe bijection with $\\mathbb{R}^2$ respects addition and multiplication by real numbers. It is the base of the [[complex plane|0CTE7A31XQ]].\nAnother construction is the quotient ring $\\mathbb{R}[X]/(X^2 + 1)$.\n\n## Explanations\n\n### L1 human\n\nA complex number is just a pair of real numbers with a special rule for multiplication.\nThe rule is made so that $(0, 1) \\cdot (0, 1) = (-1, 0)$, which says $i^2 = -1$.\n",
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