Pi Approximation Day
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07-25-2019, 01:26 PM
(This post was last modified: 07-26-2019 01:34 PM by Albert Chan.)
Post: #21
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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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