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  "body": "\n## Summary\n\nRational approximation is the approximation of functions by [[rational functions|0XDK9CGYKJ]] $p/q$ with prescribed bounds on the degrees of $p$ and $q$.\nRational functions can imitate poles, steep fronts, and singular points much better than polynomials of the same number of parameters.\nA striking case is [[Newman's theorem|JEQVZXHE6A]]: $\\abs{x}$ on $[-1, 1]$ has rational approximation error of order $e^{-c\\sqrt{n}}$, while polynomials only achieve order $1/n$.\n[[Padé approximants|1F16G4V64X]] and best uniform rational approximations are the main constructions.\n\n## Explanations\n\n### L1 human\n\nA polynomial is a sum of powers and can only grow without bound far away; it struggles to follow a function with a sharp corner or a near blow-up.\nA quotient of two polynomials has more freedom: its denominator can be small where the function is large.\nRational approximation asks how close such quotients of given degrees can come to a given function.\nFor example, $1/(1 + 25x^2)$ on $[-1, 1]$ is itself a rational function of degree $2$, but polynomials need high degree to approximate it well.\n",
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