The **Octal Numbering System** is very similar in principle to the previous hexadecimal numbering system except that in Octal, a binary number is divided up into groups of only 3 bits, with each group or set of bits having a distinct value of between 000 (0) and 111 ( 4+2+1 = 7 ).

Octal numbers therefore have a range of just “8” digits, (0, 1, 2, 3, 4, 5, 6, 7) making them a Base-8 numbering system and therefore, q is equal to “8”.

Then the main characteristics of an **Octal Numbering System** is that there are only 8 distinct counting digits from 0 to 7 with each digit having a weight or value of just 8 starting from the least significant bit (LSB). In the earlier days of computing, octal numbers and the octal numbering system was very popular for counting inputs and outputs because as it works in counts of eight, inputs and outputs were in counts of eight, a byte at a time.

As the base of an **Octal Numbers** system is 8 (base-8), which also represents the number of individual numbers used in the system, the subscript 8 is used to identify a number expressed in octal. For example, an octal number is expressed as: 237_{8}

Just like the hexadecimal system, the “octal number system” provides a convenient way of converting large binary numbers into more compact and smaller groups. However, these days the octal numbering system is used less frequently than the more popular hexadecimal numbering system and has almost disappeared as a digital base number system.

### Representation of an Octal Number

MSB | Octal Number | LSB | ||||||

8^{8} |
8^{7} |
8^{6} |
8^{5} |
8^{4} |
8^{3} |
8^{2} |
8^{1} |
8^{0} |

16M | 2M | 262k | 32k | 4k | 512 | 64 | 8 | 1 |

As the octal number system uses only eight digits (0 through 7) there are no numbers or letters used above 8, but the conversion from decimal to octal and binary to octal follows the same pattern as we have seen previously for hexadecimal.

To count above 7 in octal we need to add another column and start over again in a similar way to hexadecimal.

0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21….etc

Again do not get confused, 10 or 20 is **NOT** ten or twenty it is 1 + 0 and 2 + 0 in octal exactly the same as for hexadecimal. The relationship between binary and octal numbers is given below.

### Octal Numbers

Decimal Number | 3-bit Binary Number | Octal Number |

0 | 000 | 0 |

1 | 001 | 1 |

2 | 010 | 2 |

3 | 011 | 3 |

4 | 100 | 4 |

5 | 101 | 5 |

6 | 110 | 6 |

7 | 111 | 7 |

8 | 001 000 | 10 (1+0) |

9 | 001 001 | 11 (1+1) |

Continuing upwards in groups of three |

Then we can see that 1 octal number or digit is equivalent to 3 bits, and with two octal number, 77_{8} we can count up to 63 in decimal, with three octal numbers, 777_{8} up to 511 in decimal and with four octal numbers, 7777_{8} up to 4095 in decimal and so on.

## Octal Numbers Example No1

Using our previous binary number of 1101010111001111_{2} convert this binary number to its octal equivalent, (base-2 to base-8).

Binary Digit Value | 001101010111001111 |

Group the bits into three´s starting from the right hand side |
001 101 010 111 001 111 |

Octal Number form | 1 5 2 7 1 7_{8} |

Thus, 001101010111001111_{2} in its Binary form is equivalent to 152717_{8} in Octal form or 54,735 in denary.

## Octal Numbers Example No2

Convert the octal number 2322_{8} to its decimal number equivalent, (base-8 to base-10).

Octal Digit Value | 2322_{8} |

In polynomial form | = ( 2×8^{3} ) + ( 3×8^{2} ) + ( 2×8^{1} ) + ( 2×8^{0} ) |

Add the results | = ( 1024 ) + ( 192 ) + ( 16 ) + ( 2 ) |

Decimal number form equals: 1234_{10} |

Then, converting octal to decimal shows that 2322_{8} in its Octal form is equivalent to 1234_{10} in its Decimal form.

While **Octal** is another type of digital numbering system, it is little used these days instead the more commonly used Hexadecimal Numbering System is used as it is more flexible.