Here is a numeral system I came up with (probably independently invented by someone else before). It can only represent positive integers.

There are two symbols: unary digit (1) and separator (,).

The numbers are represented as the exponents of its prime factors in unary numeral system (base one; 0 = nothing, 1 = 1, 2 = 11, etc.), separated by commas. A number never ends with a comma, except for one (",").

Decimal | Prime factors | Prime factors (alt.) | My system |
---|---|---|---|

1 | – | 2^{0} | , |

2 | 2 | 2^{1} | 1 |

3 | 3 | 2^{0}×3^{1} | ,1 |

4 | 2^{2} | 2^{2} | 11 |

5 | 5 | 2^{0}×3^{0}×5^{1} | ,,1 |

6 | 2×3 | 2^{1}×3^{1} | 1,1 |

7 | 7 | 2^{0}×3^{0}×5^{0}×7^{1} | ,,,1 |

8 | 2^{3} | 2^{3} | 111 |

9 | 3^{2} | 3^{2} | ,11 |

10 | 2×5 | 2^{1}×3^{0}×5^{1} | 1,,1 |

13 | 13 | 2^{0}×3^{0}×5^{0}×7^{0}×11^{0}×13^{1} | ,,,,,1 |

30 | 2×3×5 | 2^{1}×3^{1}×5^{1} | 1,1,1 |

64 | 2^{6} | 2^{6} | 111111 |

Pros:

- unique representation for each positive integer
- easy primality testing (a number is prime if it contains exactly one "1" symbol)
- easy to multiply and divide (1,11,111 / ,11,1 = 1,,11 ; or in decimal: 2250 / 45 = 50)

Cons:

- difficult to add, subtract and compare (,11,,1 is followed by 111111 and then ,,1,,,1)

Last updated: 23 March 2017 | qalle at qalle dot net