### The irrational number π (pi)

The number π is defined as the ratio of the circumference of a circle to its diameter. This geometric definition does not immediately transform into numerical information. For thousands of years, people have been chasing π. The following table gives a partial history of numerical approximations to π.

Era | Approximation | Who/Where |
---|---|---|

2000 BCE | 3\(\dfrac{1}{8}\) | Mesopotania |

1200 BCE | 3 | China |

550 BCE | 3 | Old Testament |

263 BCE | 3.1459 | China |

250 BCE | Between 3\(\dfrac{10}{71}\) and 3\(\dfrac{1}{7}\) | Archimedes (Greece) |

150 BCE | 3.1416 | Ptolemy (Egypt) |

800 | To 14 decimal places | Al'Khwarizmi (Persia) |

1600 | To 17 places | Van Ceulen (Holland) |

1706 | To 100 decimal places | Machin (England) |

1853 | To 500 decimal places | Shanks (England) |

1949 | To 2000 decimal places | An early computer |

1997 | To 51 billion decimal places | Kanada (Japan) |

James Gregory (1638–75) produced one of the first mathematical formulas for the calculation of \(\pi\). This is:

\(\dfrac{\pi}{4}=1\ – \dfrac{1}{3}+\dfrac{1}{5}\ – \dfrac{1}{7}+\dfrac{1}{9}\ – \dfrac{1}{11}+\ \ldots\)

Obviously, π has fascinated people for thousands of years. It has given rise to many interesting formulas.