Decimal to Octal Conversion

To convert decimal to octal, we have to learn about both the number systems first. A number with base 8 is the octal number and a number with base 10 is the decimal number. Here we will convert a decimal number to an equivalent octal number. It is the same as converting any decimal number to binary or decimal to hexadecimal.

In decimal to binary, we divide the number by 2, in decimal to hexadecimal we divide the number by 16. In case of decimal to octal, we divide the number by 8 and write the remainders in the reverse order to get the equivalent octal number.

Decimal Number: All the numbers to the base ten are called decimal numbers. These are the commonly used numbers, which are 0-9. It has both integer part and the decimal part. It is separated by a decimal point (.). Numbers on the left of the decimal are integers and numbers on the right of the decimal is the decimal part. Example: (236.89)₁₀, (54.2)₁₀, etc.

Octal number: These are the numbers with base 8. If x is a number then the octal number is denoted as x8. It contains digits from 0 to 7. Example: (212)₈, (121)₈, etc.

Decimal

A decimal number is a number expressed in the base 10 numeral system. Decimal number's digits have 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Each digit of a decimal number counts a power of 10.

Decimal number example:

653₁₀ = 6×10² + 5×10¹ + 3×10⁰

Octal

An 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 = 1580₁₀

How to Convert from Decimal to Octal

Conversion steps:

Divide the number by 8.
Get the integer quotient for the next iteration.
Get the remainder for the octal digit.
Repeat the steps until the quotient is equal to 0.

Example

Understanding Binary-to-Octal Conversion

Binary-to-octal conversion involves converting numbers from base 2 (binary) to base 8 (octal). Binary numbers are composed of 0s and 1s, while octal numbers are composed of digits from 0 to 7.

The general approach to converting binary numbers to octal includes:

Grouping binary digits into sets of three, starting from the right (add leading zeros if necessary).
Converting each group of three binary digits to their equivalent octal digit.
Writing the octal digits in sequence to form the final octal number.

Steps for Binary-to-Octal Conversion

Step 1: Divide the binary number into groups of three digits starting from the right. Add leading zeros to complete groups if needed.

Step 2: Convert each group of three binary digits into the corresponding octal digit.

Step 3: Combine the octal digits to form the final result.

Example: Converting Binary to Octal

Convert \( 101101 \) to octal:

Group the binary digits into sets of three: \( 101101 \) becomes \( 101 \, 101 \) (no need for leading zeros).
Convert each group to octal: \( 101 \) in binary is \( 5 \) in octal.
Final octal result: \( 55 \).

Conversion Table for Quick Reference

Here is a quick reference for converting binary to octal:

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

Applications of Binary-to-Octal Conversion

Binary-to-octal conversion is commonly used in:

Computer programming and systems for simplifying binary data representation.
Digital electronics to represent data more compactly.
Networking and communication protocols.

Practice Problem

Convert \( 11001110 \) to octal:

Solution: Group the digits: \( 110 \, 011 \, 10 \). Add a leading zero to make it \( 110 \, 011 \, 010 \).
Convert each group: \( 110 → 6 \), \( 011 → 3 \), \( 010 → 2 \).
Final octal result: \( 632 \).

Binary-to-Octal Conversion Examples Table
Problem Type	Description	Steps to Solve	Example
Basic Conversion	Converting a binary number to its octal equivalent.	Group the binary digits in sets of three, starting from the right. Convert each group of three binary digits into its decimal (octal) equivalent.	For \( 110010 \): Group: \( 110 \) and \( 010 \). Convert: \( 110 = 6 \), \( 010 = 2 \). Result: \( 62 \) in octal.
Adding Leading Zeros	Ensuring the binary number has groups of three by adding leading zeros.	If the number of binary digits is not a multiple of three, add zeros to the left until it is. Follow the basic conversion process.	For \( 10101 \): Add leading zero: \( 010101 \). Group: \( 010 \) and \( 101 \). Convert: \( 010 = 2 \), \( 101 = 5 \). Result: \( 25 \) in octal.
Handling Larger Numbers	Converting longer binary numbers by grouping into threes.	Split the binary number into groups of three from the right. Convert each group independently.	For \( 111001110 \): Group: \( 111 \), \( 001 \), \( 110 \). Convert: \( 111 = 7 \), \( 001 = 1 \), \( 110 = 6 \). Result: \( 716 \) in octal.
Verifying Conversion	Checking the accuracy of binary-to-octal conversion.	Convert the octal result back to binary by replacing each octal digit with its 3-bit binary equivalent. Ensure the original binary number matches.	For \( 110010 \) (converted to \( 62 \) in octal): Convert \( 6 \) to \( 110 \) and \( 2 \) to \( 010 \). Combine: \( 110 \) and \( 010 \) to get \( 110010 \). Matches the original binary number.
Applications	Understanding where binary-to-octal conversion is used.	Octal is used in digital systems as a shorthand for binary representation. Octal makes it easier to read and write large binary numbers.	Example: Converting binary machine instructions in microprocessors to octal for easier debugging.