The conversion between binary numbers and octal numbers is an effortless and straightforward task. Let’s start.

**Key Questions:**

- Convert octal numbers into binary numbers
- Convert binary numbers into octal numbers
- Convert octal fractional numbers into binary fractional numbers
- Convert binary fractional numbers into octal fractional numbers

Octal numbers | 00 | 01 | 02 | 03 | 04 | 05 | 06 | 07 |
---|---|---|---|---|---|---|---|---|

Binary numbers | 00 | 01 | 10 | 11 | 100 | 101 | 110 | 111 |

**Binary to Octal Conversion:**

The conversion method is to make a group of three bits. But why do we make a group of 3 bits? Because 3 bits number in binary the total values can be expressed are 2^{n} or largest possible value for a 3 bit number is 2^{n-1}=7. The total digits or unique symbols in base 8 (octal) system are 8.

For integers start making groups from right to left.

**Example 01011101) _{2}**

__010__ __011__ __101__

We have a 8 bit number. Start making group of 3 bits from right. You can make 2 groups easily. For 3rd 3 bit group we have to add extra 0. As shown above

For fractional part start making group from left to right.

**Example 0.1111000111) _{2}**

0.__111__ __100__ __011__ __100__

**Example#01:110111011.11010) _{2}=?)_{2}**

Making group of 3 bits

__110__ __111__ __011__ . __110__ __100__

Binary number | 110 | 111 | 011 | . | 110 | 100 |
---|---|---|---|---|---|---|

Octal number | 6 | 7 | 3 | . | 6 | 4 |

**Answer 110111011.11010) _{2}=673.64)_{8}**

**Example#02: 101111001.010101) _{2}**

Making group of 3 bits

__101__ __111__ __001__.__010__ __101__

Binary
number | 101 | 111 | 001 | . | 010 | 101 |
---|---|---|---|---|---|---|

Octal
number | 5 | 7 | 1 | . | 2 | 5 |

**Answer 110111011.11010) _{2}=571.25)_{8}**

**Example#03: 1111110011111001) _{2}=?)_{8}**

Making group of 3 bits

__001__ __111__ __110__ __011__ __111__ __001__

*Note: add two zeros on the left in order to make group of 3 bits.*

Binary number | 001 | 111 | 110 | 011 | 111 | 001 |
---|---|---|---|---|---|---|

Octal number | 1 | 7 | 6 | 3 | 7 | 1 |

**Answer 1111110011111001) _{2}=176371)_{8}**

**Octal to Binary Conversion:**

The conversion method is to take individual octal digit and convert it in binary. Make sure convert in the way that each octal digit should represent in 3 bit binary group.

For example if

1)_{8 }=001)_{2 } right way

1)_{8 }=1)_{2} wrong way

*Note: this is wrong for this conversion technique only. Otherwise it is obviously the same. *

*1) _{8 }=1)_{2}*

**Example#01:673.64) _{8}=?)_{2 }**

Represent each octal digit in 3 bit binary number.

Octal number | 6 | 7 | 3 | . | 6 | 4 |
---|---|---|---|---|---|---|

Binary number | 110 | 111 | 011 | . | 110 | 100 |

**Answer 673.64) _{8}= 110111011.110100)_{2}**

*Or *673.64)_{8}= 110111011.1101)_{2}

*Note: you can remove the right most 0s that are unnecessary.*

**Example#02: 571.25) _{8}= ?)_{10}**

Represent each octal digit in 3 bit binary number.

Octal number | 5 | 7 | 1 | . | 2 | 5 |
---|---|---|---|---|---|---|

Binary number | 101 | 111 | 001 | . | 010 | 101 |

**Answer 571.25) _{8}= 101111001.010101)_{2}**

**Example#03: 176371) _{8}=?)_{2}**

Represent each octal digit in 3 bit binary number.

Octal number | 1 | 7 | 6 | 3 | 7 | 1 |
---|---|---|---|---|---|---|

Binary number | 001 | 111 | 110 | 011 | 111 | 001 |

**Answer 176371) _{8}= 1111110011111001)_{2}**