Pi Approximation Day

07252019, 01:26 PM
(This post was last modified: 07262019 01:34 PM by Albert Chan.)
Post: #21




RE: Pi Approximation Day
Best Pi approximation using decimal digits permutations:
85910 / 27346 = 355 / 113 ≈ 3.141592920, abs error ≈ 2.67e7 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 semiconvergent: (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 ≈ 2.94e13 

07252019, 08:14 PM
(This post was last modified: 07252019 08:35 PM by Gerson W. Barbosa.)
Post: #22




RE: Pi Approximation Day
(07242019 10:30 AM)BartDB Wrote:(07232019 06:18 PM)Gerson W. Barbosa Wrote: That is, Similar equalities can be automatically obtained on the HP50g with help of a small UserRPL program. The Egyptian Fractions part  { } WHILE SWAP DUP 5. ALOG SQ > REPEAT DUP INV CEIL DUP UNROT INV  UNROT + END 6. ALOG SQ * SWAP  is borrowed code from forumer 3298 here. For example, 22 ENTER 7 \<< / \>NUM DUP IP R\>I DUP UNROT  { } WHILE SWAP DUP 5. ALOG SQ > REPEAT DUP INV CEIL DUP UNROT INV  UNROT + END 6. ALOG SQ * SWAP NIP DUP SIZE NOT NOT { 1  X SWAP ^ 0 + \GSLIST + } { DROP } IFTE 4 X 2 ^ 1 + /  COLLECT \>> EVAL > '(X^8+X^6+3*X^21)/(X^2+1)' Indeed, '∫(0,1,(X^8+X^6+3*X^21)/(X^2+1),X)' EVAL DISTRIB > 'π+22/7' That is, π = 22/7  ∫(0,1,(X^8+X^6+3*X^21)/(X^2+1),X) Notice this is a different integrand polynomial. The original one is more elaborate so that the difference area is continuous, not distributed between both sides of the xaxis. Likewise, π = 377/120  ∫(0,1,(X^61+X^59+X^9+X^7+3*X^21)/(X^2+1),X) and π = 3 + ∫(0,1,(3*X^2+1)/(X^2+1),X) 

07262019, 11:02 PM
Post: #23




RE: Pi Approximation Day
Yeah, mine was always 355/113. I used it a lot (with 4bangers!) before calculators became available with a dedicated Pi key. It was easy to remember and pretty darned accurate.


08142019, 09:14 PM
Post: #24




RE: Pi Approximation Day
(07222019 03:14 PM)Gerson W. Barbosa Wrote: Now, time for a little riddle. There hasn’t been any response to this puzzle, but it’s my fault. As I said, I forgot to mention something else had to be assumed. I will post the answer tomorrow, but I’ll give you another tip, in case you still want to give it a try. Remember numbers are not always what they look, as in the phrase “There are 10 types of people, those who understand binary and those who don’t”. 

08142019, 09:36 PM
Post: #25




RE: Pi Approximation Day
See my latest post ‘Proving the DuffinSchaeffer conjecture’ if I’d read this one first I’d have used it as a reply instead.
Denny 

08142019, 10:46 PM
Post: #26




RE: Pi Approximation Day
(08142019 09:14 PM)Gerson W. Barbosa Wrote: \(\frac{\frac{26}{7}\frac{6}{11211}}{\left (\frac{4141}{3313}\right ) ^{\frac{3}{4}}}\) Puzzle solved If assumed all octal numbers, numerator converted back to decimal: 22/7  6/4745 = 104348/33215 = 3.141592654 (10 digits, rounded) This value happened to be one of Pi convergents, from continued fraction terms: [3;7,15,1,292,1] Thus, all is needed is to "remove" the denominator, by changing exponent to 0/4 

08142019, 11:09 PM
Post: #27




RE: Pi Approximation Day
(08142019 10:46 PM)Albert Chan Wrote:(08142019 09:14 PM)Gerson W. Barbosa Wrote: \(\frac{\frac{26}{7}\frac{6}{11211}}{\left (\frac{4141}{3313}\right ) ^{\frac{3}{4}}}\) Exactly, congrats! Perhaps I should have chosen a higher base to make it a bit more difficult to check :) 

08152019, 03:38 AM
Post: #28




RE: Pi Approximation Day
(08142019 11:09 PM)Gerson W. Barbosa Wrote: Perhaps I should have chosen a higher base to make it a bit more difficult to check :) It is even harder if the base goes negative. For negative base, we do not need the minus sign. Example, your original puzzle in negaoctal base: (166/7 + 6/172627) / (14241/15473)^{3/4} 

08202019, 04:10 PM
Post: #29




RE: Pi Approximation Day
(07262019 11:02 PM)Bill Duncan Wrote: Yeah, mine was always 355/113. I used it a lot (with 4bangers!) before calculators became available with a dedicated Pi key. It was easy to remember and pretty darned accurate. That's a really good one. (Made in China) 

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