Octal to Decimal Conversion: Step-by-Step Guide with Examples

Octal to Decimal Conversion

The octal number system is a base-8 system that uses digits from 0 to 7. It is commonly used in computing systems, especially in earlier programming practices. In contrast, the decimal system is base-10, which is the standard system for denoting integer and non-integer numbers.

Understanding Octal Numbers

• Base: 8
• Digits: 0 to 7
• Example: 157₈

Conversion Method

To convert an octal number to a decimal number, each digit is multiplied by 8 raised to the power of its position, counting from right to left, starting at 0.

Formula:

    Decimal =\((d_n \times 8^n) + (d_{n-1} \times 8^{n-1}) + ... + (d_0 \times 8^0)\)
      

Example: Convert 157₈ to Decimal

Position (right to left): 2 1 0
Octal digits: 1 5 7
Calculation:\( (1 \times 8^2) + (5 \times 8^2) + (7 \times 8^0)\)
=\( (1 \times 64) + (5 \times 8) + (7 \times 1)\)
= \(64 + 40 + 7\) = 111

Converting Octal Fractions

Octal fractions use negative powers of 8, just like decimal fractions use negative powers of 10.

Example: Convert 10.4₈ to Decimal

Integer part: \((1 \times 8^1) + (0 \times 8^0) = 8\)
Fraction part: \(4 \times 8^{-1} = 4 \times 0.125 = 0.5\)
Result: \(8 + 0.5\) = 8.5

Step-by-Step Summary

1. Write down the octal number.
2. Label each digit with its position, starting from the right (0, 1, 2, ...).
3. Multiply each digit by 8 raised to its position.
4. Add up all the results to get the decimal equivalent.

Practice Examples

• 345₈ = 229
• 71₈ = 57
• 3.2₈ = 3.25

Common Mistakes to Avoid

• Using digits greater than 7 in an octal number.
• Forgetting to start position counting from the rightmost digit.
• Mixing bases during the calculation (e.g., treating octal digits like decimal).

Applications of Octal Numbers

• File permissions in Unix/Linux: e.g., 755
• Embedded systems and firmware development
• Older programming languages and instruction sets

Octal to decimal conversion is a straightforward process once you understand positional values and base-8 multiplication. Mastery of this concept is essential for students and professionals working in computer science and electronics.