4 ms·

I did a similar thing (also with some ugly python) to generate the closest integer fraction approximations in increasing length: 3/1 = 3.0 22/7 =

I did a similar thing (also with some ugly python) to generate the closest integer fraction approximations in increasing length: 3/1 = 3.0 22/7 = 3.142857142857143 311/99 = 3.1414141414141414 355/113 = 3.1415929203539825 99733/31746 = 3.1415926415926414 312689/99532 = 3.1415926536189365 The jump in length from 355/113 to the next one is quite noticeable, and it does not happen with other irrational numbers, such as e, which progress more smoothly: 8/3 = 2.6666666666666665 19/7 = 2.7142857142857144 87/32 = 2.71875 193/71 = 2.7183098591549295 878/323 = 2.718266253869969 1457/536 = 2.718283582089552 2721/1001 = 2.7182817182817183 25946/9545 = 2.7182818229439496 49171/18089 = 2.718281828735696 271801/99990 = 2.7182818281828185

Someone 4y ago
You missed 13/4 = 3.25, 16/5 = 3.2, 19/6 = 3.16667, 179/57 = 3.14035, 201/64, 223/71, 245/78, 267/85, 289/92, 333/106, 52163/16604 https://shreevatsa.wordpress.com/2011/01/10/not-all-best-rational-approximations-are-the-convergents-of-the-continued-fraction/ https://shreevatsa.wordpress.com/2011/01/10/not-all-best-rat...
- LeoPanthera 4y ago
  I intentionally omitted fractions that were less accurate than others of equal length.
  - Someone 4y ago
    You’re discriminating against bases other than base 10 ;-) Why would the base you write the numbers down in affect whether a rational number is considered “best so far”?
    - genewitch 4y ago
      π/1, then. in base π.
    - crdrost 4y ago
      You don't need to discriminate to do it, as mentioned in comments below about continued fractions. In brief, π can be represented as a stream of integers, [ 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, 4, 2, 6, 6, 99, 1, 2, 2, 6, 3, 5, 1, 1, 6, 8, 1, 7, 1, 2, 3, 7, 1, 2, 1, 1, 12, 1, 1, 1, 3, 1, 1, 8, 1, 1, 2, 1, 6, 1, 1, 5, 2, 2, 3, 1, 2, 4, 4, 16, 1, 161, ...] To process into a rational approximation you want to maintain the two last aproximations a/b, c/d, and to get the first two numbers right you want to start with a/b = 0/1 and c/d = 1/0, zero and infinity. When you come to a number N in the sequence you form the new rational approximation (N*c + a)/(N*d + b). So you absorb 3 to generate 3/1, then you absorb 7 to generate (3×7+1)/(7×1+0)=22/7, then you absorb 15 to generate (15×22 + 3)/(15×7 + 1) = 333/106, then you absorb 1 to get this magical approximation (333 + 22)/(106 + 7) = 355/113, etc. In a sense what makes it magical is that the next number is very large. The larger it is the more decimal places you get “for free” truncating just before it. That is because this procedure is calculating the continued fraction 3 + 1/(7 + 1/(15 + 1/(1 + ...))) And you can sort of see that when we fill in the “...” with 0 to truncate the expansion there, that is not so different from truncating it with 1/292 to truncate one step later or 1/293 to truncate two steps later. Those numbers are very close to zero!
      - Someone 4y ago
        > So you absorb 3 to generate 3/1, then you absorb 7 to generate (3×7+1)/(7×1+0)=22/7, then you absorb 15 to generate (15×22 + 3)/(15×7 + 1) = 333/106 The OP did not have the sequence of best rational approximations derived from the continued fraction, but 3/1 = 3.0 22/7 = 3.142857142857143 311/99 = 3.1414141414141414 355/113 = 3.1415929203539825 That is decimal-based, picking, for any number of digits, the best (according to some criterion) approximation of pi using that number of decimal digits. They skipped 333/106, for example, because the better approximation 355/113 uses the same number of decimal digits, and included 311/99 because it happens to (just) need 5 decimal digits.
    - LeoPanthera 4y ago
      Well, OK, but I specifically wanted decimal fractions. I'm not a mathematician, from my perspective this is just a fun calculator trick. Maybe your calculator has other bases, I'm afraid mine doesn't!

Someone 4y ago

You missed 13/4 = 3.25, 16/5 = 3.2, 19/6 = 3.16667, 179/57 = 3.14035, 201/64, 223/71, 245/78, 267/85, 289/92, 333/106, 52163/16604 https://shreevatsa.wordpress.com/2011/01/10/not-all-best-rational-approximations-are-the-convergents-of-the-continued-fraction/ https://shreevatsa.wordpress.com/2011/01/10/not-all-best-rat...

LeoPanthera 4y ago
I intentionally omitted fractions that were less accurate than others of equal length.
- Someone 4y ago
  You’re discriminating against bases other than base 10 ;-) Why would the base you write the numbers down in affect whether a rational number is considered “best so far”?
  - genewitch 4y ago
    π/1, then. in base π.
  - crdrost 4y ago
    You don't need to discriminate to do it, as mentioned in comments below about continued fractions. In brief, π can be represented as a stream of integers, [ 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, 4, 2, 6, 6, 99, 1, 2, 2, 6, 3, 5, 1, 1, 6, 8, 1, 7, 1, 2, 3, 7, 1, 2, 1, 1, 12, 1, 1, 1, 3, 1, 1, 8, 1, 1, 2, 1, 6, 1, 1, 5, 2, 2, 3, 1, 2, 4, 4, 16, 1, 161, ...] To process into a rational approximation you want to maintain the two last aproximations a/b, c/d, and to get the first two numbers right you want to start with a/b = 0/1 and c/d = 1/0, zero and infinity. When you come to a number N in the sequence you form the new rational approximation (N*c + a)/(N*d + b). So you absorb 3 to generate 3/1, then you absorb 7 to generate (3×7+1)/(7×1+0)=22/7, then you absorb 15 to generate (15×22 + 3)/(15×7 + 1) = 333/106, then you absorb 1 to get this magical approximation (333 + 22)/(106 + 7) = 355/113, etc. In a sense what makes it magical is that the next number is very large. The larger it is the more decimal places you get “for free” truncating just before it. That is because this procedure is calculating the continued fraction 3 + 1/(7 + 1/(15 + 1/(1 + ...))) And you can sort of see that when we fill in the “...” with 0 to truncate the expansion there, that is not so different from truncating it with 1/292 to truncate one step later or 1/293 to truncate two steps later. Those numbers are very close to zero!
    - Someone 4y ago
      > So you absorb 3 to generate 3/1, then you absorb 7 to generate (3×7+1)/(7×1+0)=22/7, then you absorb 15 to generate (15×22 + 3)/(15×7 + 1) = 333/106 The OP did not have the sequence of best rational approximations derived from the continued fraction, but 3/1 = 3.0 22/7 = 3.142857142857143 311/99 = 3.1414141414141414 355/113 = 3.1415929203539825 That is decimal-based, picking, for any number of digits, the best (according to some criterion) approximation of pi using that number of decimal digits. They skipped 333/106, for example, because the better approximation 355/113 uses the same number of decimal digits, and included 311/99 because it happens to (just) need 5 decimal digits.
  - LeoPanthera 4y ago
    Well, OK, but I specifically wanted decimal fractions. I'm not a mathematician, from my perspective this is just a fun calculator trick. Maybe your calculator has other bases, I'm afraid mine doesn't!