However, the numbering system used in one type of circuit may be different to that of another type of circuit, for example, the memory of a computer would use hexadecimal numbers while the keyboard uses decimal numbers.

Then the conversion from one number system to another is very important with the four main forms of arithmetic being.

- Decimal – The decimal numbering system has a base of 10 (MOD-10) and uses the digits from 0 through 9 to represent a decimal number value.
- Binary – The binary numbering system has a base of 2 (MOD-2) and uses only two digits a “0” and a “1” to represent a binary number value.
- Octal – The octal numbering system has a base of 8 (MOD-8) and uses 8 digits between 0 and 7 to represent an octal number value.
- Hexadecimal – The Hexadecimal numbering system has a base of 16 (MOD-16) and uses a total of 16 numeric and alphabetic characters to represent a number value. Hexadecimal numbers consist of digits 0 through 9 and letters A to F.

Long binary numbers are difficult to both read or write and are generally converted into a system more easily understood or user friendly. The two most common derivatives based on binary numbers are the **Octal** and the **Hexadecimal** numbering systems, with both of these limited in length to a byte (8-bits) or a word (16-bits).

Octal numbers can be represented by groups of 3-bits and hexadecimal numbers by groups of 4-bits together, with this grouping of the bits being used in electronic or computer systems in displays or printouts. The grouping together of binary numbers can also be used to represent **Machine Code** used for programming instructions and control such as an **Assembly Language**.

Comparisons between the various **Decimal**, **Binary**, **Hexadecimal** and **Octal** numbers are given in the following table.

### Digital Numbering System Comparison Table

Base, b | Byte (8-bits) | Word (16-bits) |

Decimal | 0 to 255 _{10} |
0 to 65,535 _{10} |

Binary | 0000 0000 to 1111 1111 _{2} |
0000 0000 0000 0000 to 1111 1111 1111 1111 _{2} |

Hexadecimal | 00 to FF _{16} |
0000 to FFFF _{16} |

Octal | 000 to 377 _{8} |
000 000 to 177 777 _{8} |

We can see from the table above that the Hexadecimal numbering system uses only four digits to express a single 16-bit word length, and as a result it is the most commonly used **Base Numbering System** for digital, micro-electronic and computer systems.