Number Systems Conversion

In mathematics, there are different number systems to write numbers, say binary number system (base 2), octal number system (base 8), decimal (base 10), and hexadecimal number system (base 16). Octal to binary conversion is defined as converting a number from base-8 to base-2. It can be done in two ways that you will learn in this article. Let us move ahead and learn the conversion of octal to binary.

Octal

Octal number is a number expressed in the base 8 numeral system. Octal number's digits have 8 symbols: 0,1,2,3,4,5,6,7. Each digit of an octal number counts a power of 8.

Octal number example: 627₈ = 6×8² + 2×8¹ + 7×8⁰ = 407₁₀

Binary

Binary number is a number expressed in the base 2 numeral system. Binary number's digits have 2 symbols: 0 and 1. Each digit of a binary number counts a power of 2.

Binary number example: 1101₂ = 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 13₁₀

How to convert from octal to binary

To convert an octal number to a binary number, replace each octal digit with its corresponding 3-bit binary equivalent.

Example

Understanding Octal-to-Binary Conversion

Octal-to-binary conversion involves converting numbers in octal (base 8) into their equivalent binary (base 2) values. Octal uses digits 0-7 to represent values, while binary uses digits 0 and 1.

The general approach to converting octal numbers to binary includes:

Write down the octal number.
Convert each octal digit to its 3-bit binary equivalent.
Combine the binary groups to form the final binary number.

Steps for Octal-to-Binary Conversion

Step 1: Write down the octal number.

Step 2: Convert each octal digit into its 3-bit binary equivalent.

Step 3: Combine the binary groups to form the final binary number.

Example: Converting Octal to Binary

Convert \( 345 \) (octal) to binary:

Step 1: Write the octal number \( 345 \).
Step 2: Convert each digit to 3-bit binary:
- \( 3 = 011 \)
- \( 4 = 100 \)
- \( 5 = 101 \)
Step 3: Combine the binary groups: \( 011100101 \).
Final binary result: \( 011100101 \).

Conversion Table for Quick Reference

Here is a quick reference for octal digits and their binary equivalents:

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

Applications of Octal-to-Binary Conversion

Octal-to-binary conversion is commonly used in:

Simplifying binary representation in computing systems.
Interpreting file permissions in Unix/Linux systems.
Working with memory addresses and digital circuits.

Practice Problem

Convert \( 127 \) (octal) to binary:

Solution:
- \( 1 = 001 \)
- \( 2 = 010 \)
- \( 7 = 111 \)
Combine the binary groups: \( 001010111 \).
Final binary result: \( 001010111 \).

Octal-to-Binary Conversion Examples Table
Problem Type	Description	Steps to Solve	Example
Basic Conversion	Converting a single octal digit to its binary equivalent.	Write down the octal digit. Convert the octal digit to its 3-bit binary equivalent.	For `5`: ‘5’ in binary: `101`. Binary: `101`.
Handling Larger Numbers	Converting a multi-digit octal number to binary.	Convert each octal digit to its 3-bit binary equivalent. Combine the binary equivalents into one binary string.	For `275`: ‘2’ in binary: `010`. ‘7’ in binary: `111`. ‘5’ in binary: `101`. Combine: `010 111 101`. Binary: `010111101`.
Verifying Conversion	Checking the accuracy of octal-to-binary conversion.	Group the binary string into 3-bit groups from the right. Convert each 3-bit group back to its octal equivalent. Ensure the resulting octal number matches the original input.	For `010111101` (binary for `275`): Group into 3 bits: `010 111 101`. ‘010’ in octal: `2`. ‘111’ in octal: `7`. ‘101’ in octal: `5`. Octal: `275`, matches the original input.
Handling Special Cases	Converting octal numbers with leading zeroes or specific patterns to binary.	Include all leading zeroes in the binary equivalent for precision. Convert each digit individually to ensure no loss of information.	For `007`: ‘0’ in binary: `000`. ‘0’ in binary: `000`. ‘7’ in binary: `111`. Combine: `000 000 111`. Binary: `000000111`.
Applications	Understanding where octal-to-binary conversion is used.	Octal and binary are widely used in computing systems for compact representation of data. Commonly applied in permissions (e.g., file permissions in UNIX), low-level programming, and memory addressing.	Example: Converting octal permissions `754` to binary: ‘7’ in binary: `111`. ‘5’ in binary: `101`. ‘4’ in binary: `100`. Combine: `111 101 100`. Binary: `111101100`.