How to Convert from Binary to Octal

Binary and octal are two different coefficients commonly used in computers. Different from radix: binary base 2 is octal base 8, so they need to be grouped to convert. This sounds complicated, but the conversion is actually very simple.

Table of Contents

Manual transfer

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Recognize binary sequences. Binary are simple strings made up of 1’s and 0’s, such as 101001, 001, or even 1. These string literals are usually binary numbers. In addition, some books and teachers also denote binary numbers through the subscript “2”, such as 1001 ₂ , to avoid confusion with the number “one thousand and one”.

Subscript indicates the “radix” of a number. Binary is base two, octal is base 8.

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Group the 1’s and 0’s in the binary number into triplets, starting from right to left. There are eight different characters or digits used in octal and only two in binary. Because

2

3

=

8

,

{displaystyle 2^{3}=8,}

2^{3}=8,

so we need three binary digits to represent an octal digit. Group the digits from right to left. For example, the binary number 101001 would be divided into 101 001 .

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Add zero to the left of the last digit if there are not enough digits to form a triple. The number 10011011 has eight digits, and even though eight is not divisible by three, we can still convert it to octal by adding zeros first until we have enough triples. For example:

Initial number: 10011011
Group: 10 011 011
Add zero so that each group has three elements: 010 011 011 ^{[1] X Research Source}

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Add 4, 2 and 1 below each triple to note the location. Each binary number in each triple represents a position in the octal number. The first number is position 4, the second number is position 2, and the third number corresponds to position 1. For simplicity, write these numbers directly below your binary triples. For example:

010 011 011
421 421 421
001
421
110 010 001
421 421 421
Note: as a shortcut, you can skip this step and just compare the binary numbers with this octal conversion table.

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When 1 is on a positional number, write the number (4, 2 or 1) to start the octal number. If there is a 1 on “4” then your octal number has a 4. If 0 is on a positional number, your octal number will not contain that number and we will leave it blank, zero or sign strikethrough there. Consider the example problem:

Topic:
- Convert 101010011 ₂ to octal.
Group of three:
- 101 010 011
Add location numbers:
- 101 010 011
  421 421 421
Rate each position:
- 101 010 011
  421 421 421
  401 020 021 ^{[2] X Research Source}

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Add the new numbers in each triple. Once you’ve located the octal number, you just have to find the sum of the values in the triple. So with 101, we have 4, 0, 1 and get 5 (

4

+

0

+

first

=

5

{displaystyle 4+0+1=5}

4+0+1=5

). Continuing the example above:

Topic:
- Convert 101010011 ₂ to octal.
Group three, add position metrics, and rate each position:
- 101 010 011
  421 421 421
  401 020 021
Add each group of three:
- $($ $4$ $+$ $0$ $+$ $first$ $)$ $($ $0$ $+$ $2$ $+$ $0$ $)$ $($ $0$ $+$ $2$ $+$ $first$ $)$ $=$ $5$ $,$ $2$ $,$ $3$ ${displaystyle (4+0+1)(0+2+0)(0+2+1)=5,2,3}$ $(4+0+1)(0+2+0)(0+2+1)=5,2,3$

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Combine the results just obtained to form the final octal number. Breaking down the binary number makes it easier to solve the math problem – the original number was just a simple string of characters. Therefore, now, after converting, we need to combine everything to get the final result. That is all.

Topic:
- Convert 101010011 ₂ to octal.
Group of three, add the position indicator, evaluate the position and find the total:
- 101 010 011
  5 — 2 — 3
Add the numbers together:
- 523

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Add subscript 8 (like this ₈ ) to complete the conversion. Without this notation, it would be impossible to determine whether 523 is an octal or a decimal number. To let the teacher know that you have solved it correctly, add a subscript 8, indicating that it is octal, in base 8, in your answer.

Topic:
- Convert 101010011 ₂ to octal.
Convert:
- 523.
Final answer:
- 523 ₈^{[3] X Research Source}

Shortcuts and variations

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Use a simple octal conversion table to save time and homework. Although not used in testing, this is a great option for other situations. Because there are only 8 combinations of numbers in all, memorization is not difficult at all. You just need to divide the numbers into groups of three and compare them with the table in the picture. ^{[4] X Research Sources}

Note that there is no direct conversion for 8 and 9. In octal, these numbers do not exist because there are only 8 digits (0-7) in the base 8 system.

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If there is an odd part, we keep the comma and start converting from there. Consider the case of converting the binary number 10010,11 to octal. Usually, you switch from right to left and start by grouping three. With a comma, you do the conversion from that position: with the part to the left of the comma (10010), you start there and convert from right to left (010 010). With the right part (,11), you start at the comma position and convert from left to right (110). When zero is added, zero is always added in the direction of the transition. Our third group result will be 010 010 , 110.

101.1 → 101 , 100
1.01001 → 001 , 010 010
1001101,0101 → 001 001 101 , 010 100

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Use the octal conversion table to convert from octal back to binary. You need the table for the reverse conversion, because just “3” won’t give you enough information to do the math unless you already understand the octal system and want to recalculate each combination. Using the table below will make it easy to convert each octal digit to a set of three binary digits and then combine them together:

0 → 000
1 → 001
2 → 010
3 → 011
4 → 100
5 → 101
6 → 110
7 → 111 ^{[5] X Research Source}

Advice

Take the time to break down the number. Ideally, you should use large paper with plenty of room to work on.

Steps

Manual transfer

Shortcuts and variations

Advice

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