# Octal fraction to decimal

Converting octal fractions to decimal is a useful skill in computer science and digital electronics, as octal numbers (base-8) are sometimes used in these fields. This article will detail the method for converting octal fractions to decimal with thorough explanations and examples.

Table of Contents

## Understanding Octal Fractions

Octal fractions use digits from 0 to 7 and a base of 8. Each digit to the right of the octal point represents a negative power of 8, similar to how each digit to the left of the decimal point represents a positive power of 10 in the decimal system.

## Steps to Convert Octal Fractions to Decimal

To convert an octal fraction to decimal, we follow these steps:

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 1: 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, $0.526_8 \approx 0.66896875_{10}$.

### Example 2: Converting $0.74$ (Octal) to Decimal

1. Write down the octal fraction: $0.74_8$

2. Expand the octal fraction:

$0.74_8 = 7 \times 8^{-1} + 4 \times 8^{-2}$
3. Calculate the decimal values:

$7 \times 8^{-1} = 7 \times 0.125 = 0.875$ $4 \times 8^{-2} = 4 \times 0.015625 = 0.0625$
4. Sum the values:

$0.875 + 0.0625 = 0.9375$

So, $0.74_8 = 0.9375_{10}$.

## General Formula

For an octal fraction $0.d_1d_2d_3…$, the decimal equivalent can be found using the formula: $0.d_1d_2d_3…_8 = d_1 \times 8^{-1} + d_2 \times 8^{-2} + d_3 \times 8^{-3} + …$

where $d_1, d_2, d_3, …$ are the octal digits (0 to 7).

### Example 3: Converting $0.123$ (Octal) to Decimal

1. Write down the octal fraction: $0.123_8$

2. Expand the octal fraction:

$0.123_8 = 1 \times 8^{-1} + 2 \times 8^{-2} + 3 \times 8^{-3}$
3. Calculate the decimal values:

$1 \times 8^{-1} = 1 \times 0.125 = 0.125$ $2 \times 8^{-2} = 2 \times 0.015625 = 0.03125$ $3 \times 8^{-3} = 3 \times 0.001953125 = 0.005859375$
4. Sum the values:

$0.125 + 0.03125 + 0.005859375 = 0.162109375$

So, $0.123_8 \approx 0.162109375_{10}$.

## Conclusion

Converting octal fractions to decimal involves expanding the octal fraction using negative powers of 8 and summing the resulting values. This method is useful in various fields, including computer science and digital electronics. By understanding and practicing this process, you can accurately convert any octal fraction to its decimal equivalent.

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