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  "body": "\n## Definition\n\nLet $a, b, c, d$ be complex numbers with $ad - bc \\neq 0$.\nThe Möbius transformation with these coefficients is the map $f : \\widehat{\\mathbb{C}} \\to \\widehat{\\mathbb{C}}$ of the [[Riemann sphere|0CDMXGPSES]] to itself given by\n$$f(z) = \\frac{az + b}{cz + d}$$\nfor each $z \\in \\mathbb{C}$ with $cz + d \\neq 0$. The map also has the following values.\nIf $c \\neq 0$, then $f(-d/c) = \\infty$ and $f(\\infty) = a/c$.\nIf $c = 0$, then $f(\\infty) = \\infty$.\n\n## Remarks\n\nEach Möbius transformation is a bijection of $\\widehat{\\mathbb{C}}$, and its inverse is the Möbius transformation $w \\mapsto \\frac{dw - b}{-cw + a}$.\nThe Möbius transformations form a [[group|G5S1M9]] under composition, the Möbius group.\nThe coefficients $(\\lambda a, \\lambda b, \\lambda c, \\lambda d)$ with $\\lambda \\neq 0$ give the same map.\nIf $ad - bc = 0$, the formula gives a constant or is undefined, so it is not a Möbius transformation.\n\n## Explanations\n\n### L1 human\n\nA Möbius transformation is a quotient of two linear functions of $z$.\nExamples are $z \\mapsto z + 1$, $z \\mapsto 2z$ and $z \\mapsto \\frac{1}{z}$, and each Möbius transformation is a composition of maps of these simple types.\nThe point $\\infty$ is necessary, because the denominator can be zero.\n",
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