Converting Binary Fractions to Decimal: A Comprehensive Guide

Understanding how to convert binary fractions to decimal is essential in computer science and digital electronics. This guide explains the process in detail, offering a comprehensive understanding of the conversion mechanism.

Understanding Binary Fractions

Binary fractions, like their decimal counterparts, represent values between 1 and 0 but in base 2. Each bit in a binary fraction represents a power of 2, with the bit positions to the right of the binary point representing negative powers of 2, just as they do in decimal.

The Binary Fraction Representation

In a binary fraction, each bit reads as a fraction of its position, starting from 2^{-1} on the left side of the binary point. For instance, the binary fraction 0.01101_2 can be converted to decimal as follows:

2^{-2} 1/4 2^{-3} 1/8 2^{-5} 1/32

The expression 0.01101_2 translates to:

1/4 1/8 1/32 0.40625

Easier Conversion Method

A more straightforward method to convert a binary fraction to a decimal fraction is to multiply and add each binary digit by the power of 2 of its position, starting from 20 on the right side. For example, to convert 101.01 to a decimal number:

1 · 1 1 0 · 2 0 1 · 4 4 1 · 8 8 0 · 16 0

The sum is 13. When divided by 32 (25, the next higher power of 2), the result is 0.40625.

Binary Fractions: Inverse of Powers of 2

For each binary place value, the binary digit represents the inverse of 2 to the power of its position. For instance:

0.5 2^{-1} 0.25 2^{-2} 0.125 2^{-3} 0.0625 2^{-4} 0.03125 2^{-5}

The binary fraction 0.1011 can be converted to decimal as follows:

0.5 0.125 0.0625 11/16

Whole Numbers in Binary

Whole numbers in binary are straightforward to convert to decimal. For instance, 101.01_2 in binary is equal to 5.25 in decimal:

123 122 121 120 119 118 117 116 115 114 0.5 0.25 0.125 0.0625 0.03125 0.015625 0.0078125 0.00390625 0.001953125 0.0009765625

Thus, 101.01_2 5.25_{10}.

Steps for Conversion

The conversion process involves two steps:

Integral Part Conversion: Multiply each digit to the left of the binary point by the corresponding power of 2, then add the results. Fractional Part Conversion: Divide each digit to the right of the binary point by the corresponding power of 2, then add the results.

For instance, to convert 110.101_2 to decimal:

Integral Part: 1 · 1 1 1 · 2 2 0 · 4 0

The sum is 3.

Fractional Part: 1 · 1/2 0.5 0 · 1/4 0 1 · 1/8 0.125

The sum is 0.625.

Adding the results from both parts gives us 3 0.625 3.625.

Example Problems

Let's solve some example problems using the steps discussed:

Input: n 110.101 Integral Part: 6 Fractional Part: 0.625

Result: 6.625

Input: n 101.1101 Integral Part: 5 Fractional Part: 0.8125

Result: 5.8125

Conclusion

Mastering the conversion of binary fractions to decimal is fundamental for anyone involved with computer science or digital electronics. Following the systematic steps outlined in this guide, you can easily perform these conversions with confidence.