## About

A numeral system (or system of numeration) is a mathematical notation system for expressing numbers using digits (or other symbols). The numeral system gives the context that allows the (digits|symbols) to be interpreted.

This article talk about the positional notation. For other, see Non-positional positions

A positional notation has the following properties:

Example, the symbols 11 is interpreted

- as three in the binary system
- as eleven in the decimal system.

For systems for classifying numbers according to their type, see Number System (Classification|Type).

## Articles Related

## List

The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations.

Numeral System | Base | Best known prefix |
---|---|---|

Number, Numeric, Quantity | 2 | % |

Number - Octal Number System | 8 | 0 |

Number, Numeric, Quantity | 10 | Default |

Number - Hexadecimal notation (0x) | 16 | 0x |

Number - Babylonian numeral system, base-60 | 60 | ? |

### Number translation

2 | 8 | 10 | 16 |
---|---|---|---|

00000 | 0 | 0 | 0 |

00001 | 1 | 1 | 1 |

00010 | 2 | 2 | 2 |

00011 | 3 | 3 | 3 |

00100 | 4 | 4 | 4 |

00101 | 5 | 5 | 5 |

00110 | 6 | 6 | 6 |

00111 | 7 | 7 | 7 |

01000 | 10 | 8 | 8 |

01001 | 11 | 9 | 9 |

01010 | 12 | 10 | A |

01011 | 13 | 11 | B |

01100 | 14 | 12 | C |

01101 | 15 | 13 | D |

01110 | 16 | 14 | E |

01111 | 17 | 15 | F |

10000 | 20 | 16 | 10 |

10001 | 21 | 17 | 11 |

10010 | 22 | 18 | 12 |

10011 | 23 | 19 | 13 |

10100 | 24 | 20 | 14 |

10101 | 25 | 21 | 15 |

### Machine Data Representation

Most computers store data by block of bytes (of eight bits)

Representation of a full bytes 1111111 in:

Base | Representation | Note |
---|---|---|

2 | 11111111 | Eight Full Digit |

8 | 377 | Digits are not used completely (Not 777) |

10 | 255 | |

16 | FF | Digits are used completely |

As the octal digit are not used completely, the Octal system is not the most practical number system to store a byte.

## Example

Python with an set expression in order to calculate the set of numbers that are made with:

- at-most-three-digit numbers.

```
base = 2
digits = {0, 1}
{(base**2)*x + base*y + z for x in digits for y in digits for z in digits}
```

```
{0, 1, 2, 3, 4, 5, 6, 7}
```

- at-most-four-digit numbers.

`{(base**3)*w + (base**2)*x + base*y + z for x in digits for y in digits for z in digits for w in digits}`

```
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
```