The **radix** or **base** is the number of unique numerical digits, including zero, that a numeral system uses to represent numbers. For example, for the decimal system (the most common system in use today) the radix is ten, because it uses the ten digits from 0 through 9.

In any numeral system, the base will always be written as (x)_{y}. For example, (18)_{10} represents the number eighteen in the decimal system; (10)_{2} represents the number two in a base two (or binary) system. (*Radix* is a Latin word for "root". *Root* can be considered a synonym for *base* in the arithmetical sense.)

Commonly used numeral systems include:

Base | Name | Description |
---|---|---|

10 | decimal system | the most used system of numbers in the world, is used in arithmetic. Its ten digits are "0–9". Used in most mechanical counters. |

12 | duodecimal (dozenal) system | is often used due to divisibility by 2, 3, 4 and 6. It was traditionally used as part of quantities expressed in dozens and grosses. |

2 | binary numeral system | used internally by nearly all computers, is base two. The two digits are "0" and "1", expressed from switches displaying OFF and ON respectively. Used in most electric counters. |

16 | hexadecimal system | is often used in computing. The sixteen digits are "0–9" followed by "A–F". |

8 | octal system | is occasionally used in computing. The eight digits are "0–7". |

60 | sexagesimal system | originated in ancient Sumeria and passed to the Babylonians. It is still used as the basis of our modern circular coordinate system (degrees, minutes, and seconds) and time measuring (hours, minutes, and seconds). |

64 | Base 64 | is also used in computing, using as digits "A–Z", "a–z", "0–9", plus two more characters, often "+" and "/". |

256 | byte | is used internally by computers, actually grouping eight binary digits together. For reading by humans, a byte is usually shown as a pair of hexadecimal digits. |

System | 10 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

Value | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

2 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |

3 | 11 | 10 | 3 | 3 | 3 | 3 | 3 | 3 | |

4 | 100 | 11 | 10 | 4 | 4 | 4 | 4 | 4 | |

5 | 101 | 12 | 11 | 10 | 5 | 5 | 5 | 5 | |

6 | 110 | 20 | 12 | 11 | 10 | 6 | 6 | 6 | |

7 | 111 | 21 | 13 | 12 | 11 | 10 | 7 | 7 | |

8 | 1000 | 22 | 20 | 13 | 12 | 11 | 10 | 8 | |

9 | 1001 | 100 | 21 | 14 | 13 | 12 | 11 | 10 | |

10 | 1010 | 101 | 22 | 20 | 14 | 13 | 12 | 11 | |

11 | 1011 | 120 | 23 | 21 | 15 | 14 | 13 | 12 |

## Base conversion

To convert a number ‘n’ from a base system of ‘a’ to base system ‘b’, is repeatedly divided by b and all the remainders are arranged in the reverse order to get the desired number 'n' in base 'b'. Consider the following examples where ‘q’ is quotient and ‘r’ is the remainder.

Examples of base conversion

Animation : Decimal to binary