# Binary fraction to decimal

Converting binary fractions to decimal can seem complex at first, but with a clear understanding of the process, it becomes much easier. In this article, we will explore how to convert binary fractions to decimal with detailed explanations and examples.

## Understanding Binary Fractions

Binary fractions are similar to decimal fractions, but they are expressed in base-2 (binary) rather than base-10 (decimal). In a binary fraction, each digit to the right of the binary point represents a negative power of 2, just as each digit to the left represents a positive power of 2.

For example, the binary fraction $0.101$ can be expanded as follows: $0.101_2 = 1 \times 2^{-1} + 0 \times 2^{-2} + 1 \times 2^{-3}$

## Steps to Convert Binary Fraction to Decimal

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

### Example 1: Converting $0.101$ from Binary to Decimal

1. Write down the binary fraction: $0.101_2$

2. Expand the binary fraction: $0.101_2 = 1 \times 2^{-1} + 0 \times 2^{-2} + 1 \times 2^{-3}$

3. Calculate the decimal values: $1 \times 2^{-1} = 1 \times 0.5 = 0.5$ $0 \times 2^{-2} = 0 \times 0.25 = 0$ $1 \times 2^{-3} = 1 \times 0.125 = 0.125$

4. Sum the values: $0.5 + 0 + 0.125 = 0.625$

So, $0.101_2 = 0.625_{10}$.

### Example 2: Converting $0.1101$ from Binary to Decimal

1. Write down the binary fraction: $0.1101_2$

2. Expand the binary fraction: $0.1101_2 = 1 \times 2^{-1} + 1 \times 2^{-2} + 0 \times 2^{-3} + 1 \times 2^{-4}$

3. Calculate the decimal values: $1 \times 2^{-1} = 1 \times 0.5 = 0.5$ $1 \times 2^{-2} = 1 \times 0.25 = 0.25$ $0 \times 2^{-3} = 0 \times 0.125 = 0$ $1 \times 2^{-4} = 1 \times 0.0625 = 0.0625$

4. Sum the values: $0.5 + 0.25 + 0 + 0.0625 = 0.8125$

So, $0.1101_2 = 0.8125_{10}$.

### General Formula

For a binary fraction $0.b_1b_2b_3…$, the decimal equivalent can be found using the formula: $0.b_1b_2b_3… = b_1 \times 2^{-1} + b_2 \times 2^{-2} + b_3 \times 2^{-3} + …$

where $b_1, b_2, b_3, …$ are the binary digits (0 or 1).

## Conclusion

Converting binary fractions to decimal involves expanding the binary fraction using negative powers of 2 and summing the resulting values. By following the steps outlined above, you can accurately convert any binary fraction to its decimal equivalent. With practice, this process will become second nature.

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