Well this physicist doesn't. I long ago memorized pi to the number of places the computers of my younger years could handle - 3.1415926535898
But if you want a rational fraction approximation, the best one is 355/113 = 3.14159292035, an error of 0.000008491 % !
The mnemonic I used to teach my duller students was to write down the first three odd numbers twice in a string (113355) and split it down the middle to form two three digit numbers (113 355). If you can't figure out how to divide these two numbers to get a number close to three, you have no business fooling around with advanced mathematical concepts like pi.

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