Pi Approximation Day
07-25-2019, 01:26 PM (This post was last modified: 07-26-2019 01:34 PM by Albert Chan.)
Post: #21
 Albert Chan Senior Member Posts: 1,985 Joined: Jul 2018
RE: Pi Approximation Day
Best Pi approximation using decimal digits permutations:

85910 / 27346 = 355 / 113 ≈ 3.141592920, abs error ≈ 2.67e-7

Proof:

Since 355/113 is one of Pi's convergents, and the next convergent is 103993/33102
A better ratio, if exist, must be a semi-convergent: (52163+355k)/(16604+113k), k>=0

(52163+355k) + (16604+113k) (mod 9) ≡ 7 ≠ 0

Thus, better ratio does not exist.

Update: With hex digits permutations, this produce best pi approximation

0xFE86C29B / 0x5104AD73 = 4270244507 / 1359261043 ≈ 3.141592653590, abs error ≈ 2.94e-13
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