# A Trillion Pieces of Pi

The number pi (p) represents the ratio of a circle's circumference to its diameter. Starting with 3.1415926535897932384. . ., its digits run on forever. That hasn't stopped researchers from trying to calculate as many of those digits as computer technology and mathematical methods allow.

Computer scientist Yasumasa Kanada and his coworkers at the University of Tokyo Information Technology Center have now succeeded in computing 1,241,100,000,000 decimal digits of pi, smashing their own previous world record of 206,158,430,000 digits, set in 1999. The calculation required about 602 hours on a Hitachi SR8000 computer, with access to a memory of about 1 terabyte.

To calculate the digits of pi, Kanada and his team used formulas involving arctangent relations of pi. For instance, you can use the following expression to work out the value of the arctangent of *x* to any desired number of decimal places just by evaluating the series to a sufficiently large number of terms:

arctangent(*x*) = *x* – *x*^{3}/3 + *x*^{5}/5 – *x*^{7}/7 + *x*^{9}/9 – . . . .

The value of pi can then be obtained from the following equation:

p = 16 arctangent(1/5) – 4 arctangent(1/239).

By using two different formulas, the researchers were able to compare the outputs and certify the calculation's accuracy.

Improvements in the computer algorithm used for the main calculation also contributed to the feat. Kanada estimates that if the new version of the algorithm had been applied in 1999 to compute 206 billion digits of pi, the total calculation time on the same computer would have been cut from 83 to 38 hours.

The 1,241,100,000,000th decimal digit of pi (not counting the initial digit, 3) is 5. Kanada has started to analyze the statistical distribution of the digits of pi and posted preliminary results at http://www.super-computing.org/pi-decimal_current.html. The expectation is that each of the digits from 0 to 9 should appear about one-tenth of the time. In other words, you would expect the digit 7 to appear 80 billion times among the first 800 billion digits of pi. It actually occurs 79,999,775,965 times–close the expected value.

Here are Kanada's complete results for the first 800 billion digits:

Digit | Frequency |
---|---|

0 | 79,999,604,459 |

1 | 79,999,983,991 |

2 | 80,000,456,638 |

3 | 79,999,778,661 |

4 | 80,000,238,690 |

5 | 79,999,773,551 |

6 | 79,999,935,320 |

7 | 79,999,775,965 |

8 | 80,000,650,170 |

9 | 79,999,802,555 |

That's still not enough to settle questions about the distribution and apparent randomness of pi's digits. It's not even firmly established that all digits appear infinitely often, for example. No one can yet rule out the possibility that at some point beyond the range of current computations of pi's value, its decimal digits revert to a string constrained to, say, only the digits 1 and 0.