Binary to Octal

Binary and octal are both number systems used in computer science and digital electronics. Binary (base 2) is the most fundamental as it represents data using only two digits, 0 and 1. Octal (base 8) simplifies binary by grouping binary digits into sets of three, making it easier to read and interpret binary data.

This article will explain how to convert binary numbers into their octal equivalents step by step, with clear examples to illustrate the process.

Why Convert Binary to Octal?

Converting binary to octal simplifies the representation of binary data. Since one octal digit corresponds to exactly three binary digits, large binary numbers can be represented more compactly in octal, making them easier to read, debug, and write.

Binary to Octal Conversion Process

The conversion process from binary to octal involves grouping the binary digits into sets of three, starting from the right. If the number of binary digits isn’t a multiple of three, you add leading zeros to make it so. Each group of three binary digits is then converted to its octal equivalent.

Binary to Octal Conversion Table

Here is a table showing the octal equivalents for all possible three-bit binary numbers:

Binary	Octal
000	0
001	1
010	2
011	3
100	4
101	5
110	6
111	7

Step-by-Step Conversion Example

Let’s convert the binary number 110101011 to octal.

Step 1: Group the Binary Digits

Start by grouping the binary digits into sets of three, starting from the right. If necessary, add leading zeros to the leftmost group to ensure that all groups contain three digits.

Original binary number: 110101011

• Starting from the right: 011 010 101
• The leftmost group (110) already has three digits, so no leading zeros are needed.

The grouped binary number is: 110 101 011

Step 2: Convert Each Group to Octal

Next, convert each group of three binary digits to its octal equivalent using the conversion table:

• 110 in binary is 6 in octal.
• 101 in binary is 5 in octal.
• 011 in binary is 3 in octal.

Step 3: Combine the Octal Digits

Finally, combine the octal digits to get the final octal number:

• The octal equivalent of binary 110101011 is 653.

Another Example

Let’s convert the binary number 10111001 to octal.

Step 1: Group the Binary Digits

Original binary number: 10111001

• Grouping into threes from the right: 001 011 100 1
• Add leading zeros to the leftmost group to make it a full three digits: 001 011 100 001

The grouped binary number is: 001 011 100 001

Step 2: Convert Each Group to Octal

• 001 in binary is 1 in octal.
• 011 in binary is 3 in octal.
• 100 in binary is 4 in octal.
• 001 in binary is 1 in octal.

Step 3: Combine the Octal Digits

• The octal equivalent of binary 10111001 is 1341.

Practice Problem

Convert the binary number 101011 to octal.

Solution:

1. Group the binary digits: 101 011
2. Convert each group to octal:
   • 101 in binary is 5 in octal.
   • 011 in binary is 3 in octal.
3. Combine the octal digits: 53

So, the octal equivalent of binary 101011 is 53.

Conclusion

Converting binary numbers to octal is a straightforward process that simplifies the representation of binary data. By grouping binary digits into sets of three and using a simple conversion table, you can quickly and accurately convert any binary number into its octal equivalent. Understanding this process is essential for anyone working in fields related to computer science, digital electronics, or data representation.