February 04, 2003

One Hundred Interesting Mathematical Calculations, Puzzles, and Amusements, Number 12
#### Repeating Decimals

(Not a problem or a puzzle; just an amusement.)

1/1 | 1.0 |

1/2 = | 0.5 |

1/3 = | 0.[3] |

1/4 = | 0.25 |

1/5 = | 0.2 |

1/6 = | 0.1[6] |

1/7 = | 0.[142857] |

1/8 = | 0.125 |

1/9 = | 0.[1] |

1/10 = | 0.1 |

1/11 = | 0.[09] |

1/12 = | 0.08[3] |

1/13 = | 0.[076923] |

1/14 = | 0.0[714285] |

1/15 = | 0.0[6] |

1/16 = | 0.0625 |

1/17 = | 0.[0.0588235294117647] |

1/18 = | 0.0[5] |

1/19 = | 0. [052631578947368421] |

1/20 = | 0.05 |

But after doing all this work I then googled...

Posted by DeLong at February 04, 2003 08:05 PM | TrackbackEmail this entry

It's probably more an English problem than a math problem, but on this page it states

"As a matter of fact, every fraction that has an exact decimal equivalent consists only of powers of 1/2, or the product of 1/5 times a power of 1/2 or some multiple of these fractions (such as 3 times 1/4 = 3/4 = 0.75); there is no other way for an exact decimal equivalent to exist."

The key line is "product of 1/5 times a power of 1/2" which should have instead stated "a power of 1/5 times a power of 1/2". Consider 1/25 = 0.04.

An important detail, considering that it's the main point of the web page.

You missed my favorite: 80/81

Posted by: Charles on February 5, 2003 06:23 AMBen Vollmayr-Lee points out that the prime factors of ten, namely 2 and 5, matter when you consider the "does it repeat or terminate?" question. In another base, you'd get a different set of non-repeaters.

In binary fractions, for instance, even tenths are repeating. 1/10decimal = 0.0[0011]binary (hoping I did it right.) This is a peril for the novice accounting programmer, who soon learns that pennies keep disappearing if (s)he represents amounts in units of dollars.

Posted by: John Aspinall on February 5, 2003 07:09 AMIt's fairly easy to generate any repeating decimal you want-- for example,

0.345345345345... = 345/999

Homework problem: find the general formula.

Posted by: Matt on February 5, 2003 07:43 AMMatt, you're giving away the store. Call 0.345_345_ instead 115/333.

-Brian

Posted by: Brian on February 5, 2003 12:03 PMCharles:

It's my favorite too—Pat Metheny, Dewey Redman, Michael Brecker, Jack DeJohnette, and Charlie Haden!

Posted by: David on February 6, 2003 12:44 AMPost a comment