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-rat...

I intentionally omitted fractions that were less accurate than others of equal length.

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”?

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!

> 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.

π/1, then. in base π.

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!