It is a little known story—just a footnote in the calculation of π—but during the during the 17th century in England there
was considerable interest in computing decimal digits of π.
The interesting problem that circulated all across the country was not to compute more digits, but who could calculate 20 digits the most rapidly.
All sorts of people worked on this to see who would have the fastest rate.
The two sisters of the father of William Penn, noted for his involvement in Pennsylvania, were among the leaders in this enterprise.
A major topic of discussion whenever two mathematicians, professional or amateur, got together was: "What are the π rates of Penn's aunts?" |

The value

In grade school we learned that a simple approximation for

Note that all denominators between 7 and 57 (with the appropriate numerator) all give worst approximations than 22/7. As an example, the two closest possible rational values using 8 as a denominator are 25/8 = 3.12500 and 26/8 = 3.25000. Of these two, 25/8 is closer; however, neither are as good as 22/7 (3.14285...).

Of particular note is the fraction

Another rational approximation that is often used is

Successive Better Rational Approximations to π |
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numerator | denominator | approximation | error | correct digits |
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3 | / | 1 | = | 3.00000 00000 00000 . . . | − | 1.42×10^{−1} | 0 |

13 | / | 4 | = | 3.25000 00000 00000 . . . | + | 1.08×10^{−1} | 0 |

16 | / | 5 | = | 3.20000 00000 00000 . . . | + | 5.84×10^{−2} | 0 |

19 | / | 6 | = | 3.16666 66666 66666 . . . | + | 2.51×10^{−2} | 1 |

22 | / | 7 | = | 3.14285 71428 57142 . . . | + | 1.26×10^{−3} | 2 |

179 | / | 57 | = | 3.14035 08771 92982 . . . | − | 1.24×10^{−3} | 2 |

201 | / | 64 | = | 3.14062 50000 00000 . . . | − | 9.68×10^{−4} | 2 |

223 | / | 71 | = | 3.14084 50704 22535 . . . | − | 7.48×10^{−4} | 2 |

245 | / | 78 | = | 3.14102 56410 25640 . . . | − | 5.67×10^{−4} | 3 |

267 | / | 85 | = | 3.14117 64705 88235 . . . | − | 4.16×10^{−4} | 3 |

289 | / | 92 | = | 3.14130 43478 26086 . . . | − | 2.88×10^{−4} | 3 |

311 | / | 99 | = | 3.14141 41414 14141 . . . | − | 1.79×10^{−4} | 3 |

333 | / | 106 | = | 3.14150 94339 62264 . . . | − | 8.32×10^{−5} | 4 |

355 | / | 113 | = | 3.14159 29203 53982 . . . | + | 2.67×10^{−7} | 6 |

52163 | / | 16604 | = | 3.14159 23873 76535 . . . | − | 2.66×10^{−7} | 6 |

52518 | / | 16717 | = | 3.14159 23909 79242 . . . | − | 2.63×10^{−7} | 6 |

52873 | / | 16830 | = | 3.14159 23945 33570 . . . | − | 2.59×10^{−7} | 6 |

53228 | / | 16943 | = | 3.14159 23980 40488 . . . | − | 2.56×10^{−7} | 6 |

53583 | / | 17056 | = | 3.14159 24015 00937 . . . | − | 2.52×10^{−7} | 6 |

53938 | / | 17169 | = | 3.14159 24049 15836 . . . | − | 2.49×10^{−7} | 6 |

skipping 91 entries |
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86598 | / | 27565 | = | 3.14159 25993 10720 . . . | − | 5.43×10^{−8} | 6 |

86953 | / | 27678 | = | 3.14159 26006 21432 . . . | − | 5.30×10^{−8} | 7 |

87308 | / | 27791 | = | 3.14159 26019 21485 . . . | − | 5.17×10^{−8} | 7 |

skipping 42 entries |
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102573 | / | 32650 | = | 3.14159 26493 10872 . . . | − | 4.28×10^{−9} | 7 |

102928 | / | 32763 | = | 3.14159 26502 45704 . . . | − | 3.34×10^{−9} | 8 |

103283 | / | 32876 | = | 3.14159 26511 74108 . . . | − | 2.42×10^{−9} | 8 |

103638 | / | 32989 | = | 3.14159 26520 96153 . . . | − | 1.49×10^{−9} | 8 |

103993 | / | 33102 | = | 3.14159 26530 11902 . . . | − | 5.78×10^{−10} | 9 |

104348 | / | 33215 | = | 3.14159 26539 21421 . . . | + | 3.32×10^{−10} | 9 |

208341 | / | 66317 | = | 3.14159 26534 67436 . . . | − | 1.22×10^{−10} | 9 |

312689 | / | 99532 | = | 3.14159 26536 18936 . . . | + | 2.91×10^{−11} | 9 |

833719 | / | 265381 | = | 3.14159 26535 81077 . . . | − | 8.72×10^{−12} | 11 |

1146408 | / | 364913 | = | 3.14159 26535 91403 . . . | + | 1.61×10^{−12} | 10 |

3126535 | / | 995207 | = | 3.14159 26535 88650 . . . | − | 1.14×10^{−12} | 11 |

for comparison | π ≈ | 3.14159 26535 89793 |

The last three rational entries are interesting since the first of the three gives one more correct digit than the following entry. The actual error is less with the later one, but because the way the true value would round, the number of identical digits is one fewer.

The number of digits given in the table are, as normal, the number of digits to the right of the decimal point. If we examine the table we discover that for most of the rational approximations, one needs more digits combined in the numerator and denominator than are simply given in the number of correct decimal points. The exception is 355/113 for which, with the effort of memorizing these 6 digits, we get an approximation to π that contains 6 digits! It is less memorizing to simply learn the necessary decimal digits of π!