# Hexadecimal system

The **hexadecimal system** is a numeral system with base sixteen. It is the official numeral system of VetoVeto. Hexadecimal can be shortened to hex.

## Representation

The hexadecimal system uses sixteen distinct symbols to represent the numerals. Nowadays, it is common practice to use 0 to 9 and A to F to count to fifteen in hexadecimal. Decimal numerals are denoted with a subscript 10, whereas hexadecimal numerals are denoted with a subscript 16. For instance, 10_{10} corresponds to A_{16}. See the table for more examples.

Decimal | Hexadecimal |
---|---|

1_{10} |
1_{16} |

2_{10} |
2_{16} |

2_{10} |
2_{16} |

3_{10} |
3_{16} |

4_{10} |
4_{16} |

5_{10} |
5_{16} |

6_{10} |
6_{16} |

7_{10} |
7_{16} |

8_{10} |
8_{16} |

9_{10} |
9_{16} |

10_{10} |
A_{16} |

11_{10} |
B_{16} |

12_{10} |
C_{16} |

13_{10} |
D_{16} |

14_{10} |
E_{16} |

15_{10} |
F_{16} |

16_{10} |
10_{16} |

32_{10} |
20_{16} |

Negative hexadecimal numbers work similar to negative decimal numbers. For instance, -10_{10} corresponds to -A_{16}. Rules regarding multiplication of negative numbers are not different either, e.g. -5_{10} × 2_{10} = -10_{10} = -A_{16}.

## Hexadecimal fractions

As the hexadecimal numeral system uses base sixteen that does not easily translate to base 10, several decimal fractions do not have a simple hexadecimal representation and vice versa. In base ten we can easily divide each number in tenths: 0.1_{10}, 0.2_{10} and so on. Conversely, we divide each number in sixteenths in hexadecimal: 0.1_{16} = 1/16_{10} = 0.0625_{16}. A half is five tenths in decimal, thus 5/10_{10} = 0.5_{10}. A half is 8 sixteenths in hexadecimal, thus 8/10_{16} = 0.8_{16}. Nine tenths is 9/10_{10} = 0.9_{10}, but this 0.E66..._{16} in hexadecimal, where 6 is a repeating numeral.

In fact, most decimal fractions have a corresponding hexadecimal fraction with a repeating numeral. For instance, 0.1_{10} corresponds to 0.199..._{16} where 9 is a repeating numeral.

## Use in languages

The hexadecimal system is used in computing as there is a one-to-one correspondence between one hexadecimal digit and four binary digits. Besides applications in computing, the hexadecimal system does not see much use in everyday spoken and written languages of Earth, however.

Hexadecimal is the official counting system in the vast majority of languages of VetoVeto, such as Qromean and Sygvaraæn.