[ Home -- Blog -- Reference -- About ] Permutation Fractions --------------------- 355/133 is a famous fractional apporo- ximation of PI. It is roughly equal to 3.1415929.., extremely close to the true value 3.141592653589... However, it only uses three of the available decimal di- gits, and it uses them three times each! How close can we get if we restrict both the numerator and denominator to use all digits 1-9, one time each? Given that there are 9! = 352880 possib- le numerators, and as many denominators, this is very brute-forcable. A quick and dirty python script was able to calcu- late optimal fractions using the digits 1 to 7 in a few seconds, 1 to 8 took a few minutes, and letting it run over- night, took care of the final case (for no reason at all, I didn't think of inc- luding zero in any of this). Results ------- Digs | Best fraction | PI 1 | 1 / 1 | 1.0 1-2 | 21 / 12 | 1.75 1-3 | 321 / 123 | 2.60... 1-4 | 4213 / 1342 | 3.139... 1-5 | 42531 / 13542 | 3.1406... 1-6 | 516243 / 164325 | 3.141597... 1-7 | 6734215 / 2143567 | 3.1415929... Using digits 1-8 gives the optimal frac- tion: 87435126 / 27831465 = 3.1415926542... with an impressive eight correct decimal places! And finally, all digits 1-9 gives: 467895213 / 148935672 = 3.14159265350... with ten correct decimals. Addendum -------- A similar constraint is to only use each digit once, across both the numerator and the denominator. This gives a much simpler brute force problem, so solving that took my dirty python code no time at all. This time I even remembered to include zero! The optimal fraction in this case is: 85910 / 27346 = 3.1415929203... Coincidentally, this actually reduces to the "standard" fractional approximation, 355/113! Use this information for whatever purpo- se you might find for it![ Home -- Blog -- Reference -- About ] ---------------------------------------- 99v.no (c)2026v > Discussion
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