I am planning on running a half marathon next month. It is not my first, but it has been a while since I have done one. In 2011, my nephew designed a "Training for a Half Marathon" t-shirt that used a binary representation of the distance. I checked the math. He was working for my dad when he did this. My dad, Ken Long, organized running events and training classes to help people prepare for different running events before he retired in 2013. I thought it would be interesting to look at the distances in a variety of different base counting systems. I may add more later, but for now, I am just using base 2 through base 10.

First of all, you need to know that a full marathon is 26 miles and 385 yards, so a half marathon is 13 miles and 192.5 yards. There are 1760 yards in a mile, so a half marathon is 13 + 192.5/1760 = 13 + 385/3520 = 13 + 7/64 miles.

Base n |
13 converted to base n |
7/64 converted to base n |
decimal representation in base n |
---|---|---|---|

2 | 1101 | 111/1,000,000 | 1101.000111 |

3 | 111 | 21/2101 | 111.0022212012111001 |

4 | 31 | 13/1000 | 31.013 |

5 | 23 | 12/224 | 23.0233134430104122 |

6 | 21 | 11/144 | 21.035343 |

7 | 16 | 10/121 | 16.05234160 |

8 | 15 | 7/100 | 15.07 |

9 | 14 | 7/71 | 14.08765432 |

10 | 13 | 7/64 | 13.109375 |

I am a math professor, but anybody can make mistakes. Let me know if something doesn't seem right. I derived all of these answers, and I checked them. Checking the ones with repeating decimals was quite a bit of work especially when the numbers got so big that my calculator wanted to start to round off. I used hand calculations when I needed to and I believe these all to be correct and I may check them again after I improve one of my base converting programs to be able to handle decimals.

Last Update: April 7, 2015