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 numbers' 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 numbers' 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 Decimal-to-Octal Conversion

Decimal-to-octal conversion involves converting decimal numbers (base 10) into their corresponding octal representations (base 8). Octal uses digits 0-7 to represent values from 0 to 7, respectively.

The general approach to converting decimal numbers to octal includes:

Divide the decimal number by 8 and record the remainder.
Repeat the division until the quotient is 0.
Write the remainders in reverse order to get the octal representation.

Steps for Decimal-to-Octal Conversion

Step 1: Divide the decimal number by 8, noting the quotient and remainder.

Step 2: Repeat the division until the quotient is 0.

Step 3: Write the remainders in reverse order to obtain the octal number.

Example: Converting Decimal to Octal

Convert \( 254 \) to octal:

Step 1: Divide \( 254 \) by \( 8 \): quotient \( 31 \), remainder \( 6 \).
Step 2: Divide \( 31 \) by \( 8 \): quotient \( 3 \), remainder \( 7 \).
Step 3: Divide \( 3 \) by \( 8 \): quotient \( 0 \), remainder \( 3 \).
Write the remainders in reverse order: \( 376 \).
Final octal result: \( 376 \).

Conversion Table for Quick Reference

Here is a quick reference for some decimal numbers and their octal equivalents:

0 → 0
1 → 1
2 → 2
5 → 5
8 → 10
15 → 17
16 → 20
64 → 100
255 → 377

Applications of Decimal-to-Octal Conversion

Decimal-to-octal conversion is commonly used in:

Representing numbers in computing systems, especially in early computer systems for memory addresses and file permissions.
Compact representation of binary data in programming and digital systems.
Working with low-level computing tasks, such as in UNIX-based systems for file permissions.

Practice Problem

Convert \( 103 \) to octal:

Solution:
- Step 1: Divide \( 103 \) by \( 8 \): quotient \( 12 \), remainder \( 7 \).
- Step 2: Divide \( 12 \) by \( 8 \): quotient \( 1 \), remainder \( 4 \).
- Step 3: Divide \( 1 \) by \( 8 \): quotient \( 0 \), remainder \( 1 \).
Write the remainders in reverse order: \( 147 \).
Final octal result: \( 147 \).

Decimal-to-Octal Conversion Examples Table
Problem Type	Description	Steps to Solve	Example
Basic Conversion	Converting a decimal number to its octal equivalent.	Divide the decimal number by 8. Record the remainder. Repeat the process with the quotient until the quotient is 0. Write the remainders in reverse order to get the octal result.	For \( 72 \): Divide by 8: \( 72 \div 8 = 9 \) remainder \( 0 \). Divide by 8: \( 9 \div 8 = 1 \) remainder \( 1 \). Divide by 8: \( 1 \div 8 = 0 \) remainder \( 1 \). Octal: \( 110 \).
Handling Larger Numbers	Converting larger decimal numbers to octal.	Follow the same division-by-8 method, recording each remainder. Write the remainders in reverse order.	For \( 255 \): Divide by 8: \( 255 \div 8 = 31 \) remainder \( 7 \). Divide by 8: \( 31 \div 8 = 3 \) remainder \( 7 \). Divide by 8: \( 3 \div 8 = 0 \) remainder \( 3 \). Octal: \( 377 \).
Verifying Conversion	Checking the accuracy of decimal-to-octal conversion.	Convert the octal number back to decimal. Ensure the decimal result matches the original number.	For \( 110 \) (octal for \( 72 \)): Octal: \( 110 \). Convert back to decimal: \( 1 \times 8^2 + 1 \times 8^1 + 0 \times 8^0 = 72 \). Decimal: \( 72 \), matches the original input.
Handling Special Cases	Converting special decimal numbers to octal (e.g., powers of 8).	For powers of 8, the octal representation will have one digit and powers of 8.	For \( 64 \): Decimal: \( 64 \). Octal: \( 100 \).
Applications	Understanding where decimal-to-octal conversion is used.	Octal representation is used in computing systems, especially in early computer systems for memory addressing and file permissions (e.g., in UNIX-based systems). Octal is also used in certain network communication protocols and other technical fields.	Example: Converting decimal numbers to octal for use in file permissions in UNIX systems.