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  "body": "\n## Description\n\nA complex number is a number of the form $a + b i$, where $a$ and $b$ are [[real numbers|08J2M1JYAG]] and $i$ is the [[imaginary unit|0ESCX36MMC]], with $i^2 = -1$.\nWe write $\\mathbb{C}$ for the set of complex numbers.\nThe real number $a$ is the real part and $b$ is the imaginary part of $a + b i$, see [[real and imaginary parts|02ZE6JHJ7G]].\nAddition and multiplication follow the usual rules of algebra together with $i^2 = -1$:\n$$(a + b i) + (c + d i) = (a + c) + (b + d) i, \\qquad (a + b i)(c + d i) = (a c - b d) + (a d + b c) i.$$\nWith these operations, $\\mathbb{C}$ is a [[field|0FRSR10DWF]] that contains $\\mathbb{R}$ as the numbers $a + 0 i$.\n\n## Remarks\n\nWe picture $\\mathbb{C}$ as the [[complex plane|0CTE7A31XQ]], with the point $(a, b)$ for $a + b i$.\nEvery polynomial equation of positive degree with complex coefficients has a solution in $\\mathbb{C}$. This is the fundamental theorem of algebra.\nUnlike $\\mathbb{R}$, the field $\\mathbb{C}$ [[cannot be ordered|01CJRJKDAA]].\n\n## Explanations\n\n### L1 human\n\nThe equation $x^2 = -1$ has no real solution, because a real square is never negative.\nThe complex numbers add a new number $i$ with $i^2 = -1$, and then all the numbers $a + b i$ that you can build from it.\nFor example, $(1 + 2 i)(3 - i) = 3 - i + 6 i - 2 i^2 = 5 + 5 i$.\n",
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