Octal to Decimal

Converting octal numbers to decimal is an important concept in computer science and digital electronics, as octal numbers (base-8) are often used in programming and digital systems. In this article, we will explore how to convert both whole octal numbers and octal fractions to decimal, with detailed explanations and examples.

Understanding Octal Numbers

Octal numbers use digits from 0 to 7. Each digit in an octal number represents a power of 8, similar to how each digit in a decimal number represents a power of 10.

Converting Whole Octal Numbers to Decimal

To convert a whole octal number to decimal, we use the positional value of each digit. Here’s the step-by-step process:

1. Write down the octal number.
2. Expand the octal number using powers of 8. Each digit is multiplied by $8^n$, where $n$ is the position of the digit from right to left, starting at 0.
3. Sum the expanded values. This sum will be the decimal equivalent of the octal number.

Example 1: Converting 745 (Octal) to Decimal

1. Write down the octal number: $745_8$

2. Expand the octal number:

   $$745_8 = 7 \times 8^2 + 4 \times 8^1 + 5 \times 8^0$$

3. Calculate the decimal values:

   $$7 \times 8^2 = 7 \times 64 = 448$$ $$4 \times 8^1 = 4 \times 8 = 32$$ $$5 \times 8^0 = 5 \times 1 = 5$$

4. Sum the values:

   $$448 + 32 + 5 = 485$$

So, $745_8 = 485_{10}$.

Converting Octal Fractions to Decimal

To convert an octal fraction to decimal, we use the positional value of each digit, similar to whole numbers but with negative powers of 8. Here’s how it works:

1. Write down the octal fraction.
2. Expand the octal fraction using powers of 8. Each digit to the right of the octal point is multiplied by $8^{-n}$, where $n$ is the position of the digit, starting at 1.
3. Sum the expanded values. This sum will be the decimal equivalent of the octal fraction.

Example 2: Converting 0.526 (Octal) to Decimal

1. Write down the octal fraction: $0.526_8$

2. Expand the octal fraction:

   $$0.526_8 = 5 \times 8^{-1} + 2 \times 8^{-2} + 6 \times 8^{-3}$$

3. Calculate the decimal values:

   $$5 \times 8^{-1} = 5 \times 0.125 = 0.625$$ $$2 \times 8^{-2} = 2 \times 0.015625 = 0.03125$$ $$6 \times 8^{-3} = 6 \times 0.001953125 = 0.01171875$$

4. Sum the values:

   $$0.625 + 0.03125 + 0.01171875 = 0.66896875$$

So, $0.526_8 \approx 0.66896875_{10}$.

Converting Mixed Octal Numbers to Decimal

For numbers with both whole and fractional parts, convert each part separately and then combine the results.

Example 3: Converting 745.526 (Octal) to Decimal

1. Convert the whole number part (745) to decimal:

   $$745_8 = 7 \times 8^2 + 4 \times 8^1 + 5 \times 8^0 = 448 + 32 + 5 = 485$$

2. Convert the fractional part (0.526) to decimal:

   $$0.526_8 = 5 \times 8^{-1} + 2 \times 8^{-2} + 6 \times 8^{-3} = 0.625 + 0.03125 + 0.01171875 \approx 0.66896875$$

Combine the two parts to get the decimal representation: $745.526_8 \approx 485.66896875_{10}$

Conclusion

Converting octal numbers to decimal involves using the positional values of the digits, with each digit multiplied by the appropriate power of 8. This method applies to both whole numbers and fractions. Understanding this process is crucial for working with systems that use octal numbering, such as certain digital and computing applications. With practice, you can accurately convert any octal number to its decimal equivalent.