π is an irrational number, which means that it cannot be expressed as a simple fraction such as 1/5 or 3/4. However, you can get pretty close.

A fraction that is often used is 22/7. This is not all that good:

22/7 | = | 3.14285714... |

π | = | 3.14159265... |

Although within 0.04% of the correct answer, 22/7 is only correct to 2 decimal places. We can do better than this.

355/113 is correct to 6 decimal places. It's within 0.000009% of π.

355/113 | = | 3.1415929203... |

π | = | 3.1415926535... |

355/113 is such a good approximation to π, that there is not a more accurate fraction until 52163 / 16604, and that is only marginally closer to π, still only correct to 6 decimal places.

52163/16604 | = | 3.1415923874... |

355/113 | = | 3.1415929203... |

π | = | 3.1415926535... |

To be accurate to 7 decimal places we need to go as far as 86953 / 27678

86953/27678 | = | 3.1415926006... |

52163/16604 | = | 3.1415923874... |

355/113 | = | 3.1415929203... |

π | = | 3.1415926535... |

The importance of 355/113 has been recognized, giving it the name Milü.

There's an easy way to remember 355/113:

ReplyDeleteWrite down each number twice: 1 1 3 3 5 5

That's 1,3,5, of course.

split the sequence in the middle:

113 355

If you have Matlab, you can see how absurdly accurate 355/113 is:

ReplyDeleteformat long;

bestapproxerror = .2;

tic

a=[];

for i=1:100000

for j=1:round(i/3)

if(abs((i/j)-pi)<bestapproxerror)

bestapproxerror = abs((i/j)-pi);

a(size(a,1)+1,:) = [i j bestapproxerror toc]

tic

end

end

end

plot(a(:,1),a(:,3))