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All Rational Approximations of Pi Are Useless  http://blog.wolfram.com/2011/06/30/allrationalapproximation...
355/113 is a bit better than others. They speak about relative efficiency, while absolute size is also important sometimes.

9000
I have solved the Pi memorization problem with the help of my cell phone provider. My phone number starts with 6 digits that come after 3.1415 so I know Pi with much more precision that any reasonable rational approximation can yield.

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