Binary Fractions and Negative Powers of Two

In computing and mathematics, fractional quantities in the binary number system are represented using negative powers of two placed to the right of a binary point (also known as a radix point). Just as the decimal system uses negative powers of ten (\(10^{-1}, 10^{-2}, 10^{-3}\)) to represent tenths, hundredths, and thousandths, the binary system uses negative powers of two (\(2^{-1}, 2^{-2}, 2^{-3}\)) to represent halves, quarters, and eighths. This article explains how negative exponents function in binary positional notation, how to calculate binary fractions, and the fundamental mechanics of converting between binary and decimal fractional values.

The Binary Radix Point and Positional Values

Positional notation determines the value of a digit based on its position relative to a base point. In binary, digits to the left of the radix point represent whole numbers using non-negative powers of two (\(2^0, 2^1, 2^2, \dots\)). Digits to the right of the radix point represent fractional parts using negative powers of two.

The place values decrease by a factor of two with each step to the right:

• \(2^{-1} = \frac{1}{2^1} = \frac{1}{2} = 0.5\)
• \(2^{-2} = \frac{1}{2^2} = \frac{1}{4} = 0.25\)
• \(2^{-3} = \frac{1}{2^3} = \frac{1}{8} = 0.125\)
• \(2^{-4} = \frac{1}{2^4} = \frac{1}{16} = 0.0625\)
• \(2^{-5} = \frac{1}{2^5} = \frac{1}{32} = 0.03125\)

Calculating Binary Fractions

To evaluate a binary fraction, you multiply each bit (either \(0\) or \(1\)) by its corresponding negative power of two and sum the results.

Example: Converting \(0.1101_2\) to Decimal

Given the binary fraction \(0.1101_2\):

\[\text{Value} = (1 \times 2^{-1}) + (1 \times 2^{-2}) + (0 \times 2^{-3}) + (1 \times 2^{-4})\]

Substitute the decimal equivalents:

\[\text{Value} = (1 \times 0.5) + (1 \times 0.25) + (0 \times 0.125) + (1 \times 0.0625)\] \[\text{Value} = 0.5 + 0.25 + 0 + 0.0625 = 0.8125_{10}\]

Thus, \(0.1101_2\) equals \(0.8125_{10}\).

Converting Decimal Fractions to Binary

To convert a decimal fraction into its binary representation, use the repeated multiplication by 2 method:

1. Multiply the fractional part by \(2\).
2. The integer part of the result (\(0\) or \(1\)) becomes the next binary digit.
3. Take the remaining fractional part and repeat the process until the fraction becomes \(0\) or the required precision is reached.

Example: Converting \(0.625_{10}\) to Binary

• \(0.625 \times 2 = \mathbf{1}.25\) (Record 1, keep \(0.25\))
• \(0.25 \times 2 = \mathbf{0}.50\) (Record 0, keep \(0.5\))
• \(0.5 \times 2 = \mathbf{1}.00\) (Record 1, fraction is now \(0\))

Reading the recorded integer values from top to bottom yields \(0.101_2\).

Limitations and Repeating Binary Fractions

A decimal fraction can only be represented as a finite (terminating) binary fraction if its denominator in lowest terms is a power of two. Fractional values such as \(0.1_{10}\) (\(\frac{1}{10}\)) or \(0.2_{10}\) (\(\frac{1}{5}\)) cannot be expressed precisely with a finite sum of negative powers of two. Instead, they result in infinite repeating binary sequences (for example, \(0.1_{10} = 0.0001100110011\dots_2\)), which is the underlying cause of rounding inaccuracies in computer floating-point arithmetic.