Table of Contents

Last modified on October 20th, 2023

The octal number system is the base 8 number system. Thus, it has 8 characters or numbers starting from 0 to 8. It is written as 214_{8}.

Some more examples of octal numbers are 558, 1018, 608

In 1801, James Anderson suggested the base 8 number system and coined the term ‘octal.’

Like the hexadecimal number system, the advantage of the octal number system is that it provides an easier way of converting large binary numbers into more compact and smaller groups. However, it is less commonly used due to its limitations of having elements till 7.

Although it is widely used in computer applications and other digital systems, computers cannot comprehend octal numbers. Thus, they convert it to binary numbers first.

To convert octal numbers to binary, we convert each number from the octal number to the binary number.

Here is the table of the octal numbers with their corresponding binary equivalents. We use 3 bits to represent all octal numbers. Each group has a value between 000 and 111. Thus, numbers above 7, like 11, 12, and 19 are not octal.

Octal | Binary |
---|---|

0 | 000 |

1 | 001 |

2 | 010 |

3 | 011 |

4 | 100 |

5 | 101 |

6 | 110 |

7 | 111 |

Let us convert (15)_{8} into the binary number system.

The given octal number is (18)_{8}

Using the octal-to-binary conversion table, we get

1_{8} →001_{2}

5 →101_{2}

Thus, (15)_{8 }→ (001101)_{2}

To convert an octal number to a decimal number, it is expanded with the base of eight, each digit multiplied by the power of 8, based on its position.

Let us convert (282)_{8} into the decimal number system.

The given octal number is (282)_{8}

To convert it to the decimal form:

(282)_{8} = (2 × 8^{2}) + (2 × 8^{1}) + (2 × 8^{0})

= 128 + 16 + 2

= 146

Thus, (282)_{8 }→ (146)_{10}

Check our hexadecimal number system article to learn this conversion.

Converting a binary number to an octal number is similar to an octal-to-binary conversion using the table.

Let us convert (101011)_{2} to an octal number.

The given binary number is (101011)_{2}

To convert it to the corresponding octal number, we will use the table below:

Octal | Binary |
---|---|

0 | 000 |

1 | 001 |

2 | 010 |

3 | 011 |

4 | 100 |

5 | 101 |

6 | 110 |

7 | 111 |

101 → 5

011 → 3

Thus, (101011)_{2}→ (53)_{2}

Here, the decimal number is divided by 8 with the reminder obtained from the previous division.

Let us convert (712)_{10} to an octal number

Division | Quotient | Remainder |
---|---|---|

712 ÷ 8 | 89 | 0 |

89 ÷ 8 | 11 | 1 |

11 ÷ 8 | 1 | 3 |

1 ÷ 8 | 0 | 1 |

Thus,

(712)_{10} → (1310)_{8}

Check our hexadecimal number system article to learn this conversion.

Like decimal numbers, octal numbers can be multiplied. One way is to convert the octal number to a decimal number, multiply the decimal numbers, and then convert the product to octal.

The other way is to use the octal multiplication table given:

× | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

2 | 0 | 2 | 4 | 6 | 10 | 12 | 14 | 16 |

3 | 0 | 3 | 6 | 11 | 14 | 17 | 22 | 25 |

4 | 0 | 4 | 10 | 14 | 20 | 24 | 30 | 34 |

5 | 0 | 5 | 12 | 17 | 24 | 31 | 36 | 43 |

6 | 0 | 6 | 14 | 22 | 30 | 36 | 44 | 52 |

7 | 0 | 7 | 16 | 25 | 34 | 43 | 52 | 61 |