The *hexadecimal* or base-16 numbering system is commonly used as a condensed representation of a binary number, since each hexadecimal digit is equivalent to four binary digits.

The main characteristics of the hexadecimal numbering system are that:-

- each digit varies in the range 0 to 15,
- moving to the left, each digit is worth sixteen times as much as the digit immediately to its right.

The octal or base-8 numbering system is an alternative to hexadecimal, in which each octal character is equivalent to three binary digits.

A single hexadecimal digit may take any value between 0 and 15. Numbers 0–9 are used as in the decimal system, while values in the range 10–15 are represented by letters of the alphabet A, B, C, D, E and F, as shown in the following table.

Bit 3 (8) |
Bit 2 (4) |
Bit 1 (2) |
Bit 0 (1) |
Hexadecimal Value |
---|---|---|---|---|

0 | 0 | 0 | 0 | 0 |

0 | 0 | 0 | 1 | 1 |

0 | 0 | 1 | 0 | 2 |

0 | 0 | 1 | 1 | 3 |

0 | 1 | 0 | 0 | 4 |

0 | 1 | 0 | 1 | 5 |

0 | 1 | 1 | 0 | 6 |

0 | 1 | 1 | 1 | 7 |

1 | 0 | 0 | 0 | 8 |

1 | 0 | 0 | 1 | 9 |

1 | 0 | 1 | 0 | A |

1 | 0 | 1 | 1 | B |

1 | 1 | 0 | 0 | C |

1 | 1 | 0 | 1 | D |

1 | 1 | 1 | 0 | E |

1 | 1 | 1 | 1 | F |

## Converting Between Binary and Hexadecimal

Conversion between binary and hexadecimal is based on swapping groups of 4-bits with the equivalent hexadecimal digit. Unfortunately, there is no easy way to remember the associated 4 bit binary codes for the letters A to F, other than regular practice. However, the above table may easily be drawn from first principles, if required.

#### Binary to Hexadecimal Conversion

When converting between the binary and hexadecimal numbering systems, each group of 4 bits starting at the right hand side is converted into a single hexadecimal digit. This approach is illustrated by the following example, which converts the binary number 111 1010 1100 1110_{2} giving the result 7ACE_{16}.

#### Hexadecimal to Binary Conversion

To convert a hexadecimal number to binary, each hexadecimal digit is replaced by the equivalent 4-bit code. The following example converts the hexadecimal number 4FD7_{16} to binary, giving the result 100 1111 1101 0111_{2}.

See the Numbering Systems Problems page for additional practice activities. Windows users may check their results by using the Windows Calculator in programmer’s view.