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RE: Pi Approximation Day - Albert Chan - 07-25-2019 01:26 PM
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
RE: Pi Approximation Day - Gerson W. Barbosa - 07-25-2019 08:14 PM
(07-24-2019 10:30 AM)BartDB Wrote:(07-23-2019 06:18 PM)Gerson W. Barbosa Wrote: That is,
π = 22/7 - ∫(0,1,X^4*(1-X)^4/(1+X^2),X)
Similar equalities can be automatically obtained on the HP-50g with help of a small User-RPL 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^2-1)/(X^2+1)'
Indeed,
'∫(0,1,(X^8+X^6+3*X^2-1)/(X^2+1),X)'
EVAL DISTRIB
--> '-π+22/7'
That is,
π = 22/7 - ∫(0,1,(X^8+X^6+3*X^2-1)/(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 x-axis.
Likewise,
π = 377/120 - ∫(0,1,(X^61+X^59+X^9+X^7+3*X^2-1)/(X^2+1),X)
and
π = 3 + ∫(0,1,(-3*X^2+1)/(X^2+1),X)
RE: Pi Approximation Day - Bill Duncan - 07-26-2019 11:02 PM
Yeah, mine was always 355/113. I used it a lot (with 4-bangers!) before calculators became available with a dedicated Pi key. It was easy to remember and pretty darned accurate.
RE: Pi Approximation Day - Gerson W. Barbosa - 08-14-2019 09:14 PM
(07-22-2019 03:14 PM)Gerson W. Barbosa Wrote: Now, time for a little riddle.
The following appears to be a pretty bad approximation. It really is, depending on how we look at it. However, when I change only one digit or, equivalently, when I remove one of its parts, it returns a perfect 10-digit result on my HP-41C, which I am using to evaluate it. BTW, I have used the HP-41C for this one because of its 10-digit display and a seldom used useful built-in function which most of my Voyagers lack. Too many tips, but it doesn't matter :-)
Have fun!
\(\frac{\frac{26}{7}-\frac{6}{11211}}{\left (\frac{4141}{3313}\right ) ^{\frac{3}{4}}}\)
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”.
RE: Pi Approximation Day - Leviset - 08-14-2019 09:36 PM
See my latest post ‘Proving the Duffin-Schaeffer conjecture’- if I’d read this one first I’d have used it as a reply instead.
RE: Pi Approximation Day - Albert Chan - 08-14-2019 10:46 PM
(08-14-2019 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
RE: Pi Approximation Day - Gerson W. Barbosa - 08-14-2019 11:09 PM
(08-14-2019 10:46 PM)Albert Chan Wrote:(08-14-2019 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
Exactly, congrats!
Perhaps I should have chosen a higher base to make it a bit more difficult to check :-)
RE: Pi Approximation Day - Albert Chan - 08-15-2019 03:38 AM
(08-14-2019 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
RE: Pi Approximation Day - Gerson W. Barbosa - 08-20-2019 04:10 PM
(07-26-2019 11:02 PM)Bill Duncan Wrote: Yeah, mine was always 355/113. I used it a lot (with 4-bangers!) 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)
![[Image: 48389175122_ac075da348_b.jpg]](https://live.staticflickr.com/65535/48389175122_ac075da348_b.jpg)
![[Image: 48389175192_ac075da348_b.jpg]](https://live.staticflickr.com/65535/48389175192_ac075da348_b.jpg)