Binary Fraction Converter

What Is a Binary Fraction?

A binary fraction represents the fractional or non-integer portion of a number in base-2 (binary) positional notation. While standard decimal fractions express values using negative powers of 10 ($10^{-1} = 0.1$, $10^{-2} = 0.01$, $10^{-3} = 0.001$), binary fractions express fractional magnitudes as sums of negative powers of 2:

$$2^{-1} = \frac{1}{2} = 0.5, \quad 2^{-2} = \frac{1}{4} = 0.25, \quad 2^{-3} = \frac{1}{8} = 0.125, \quad 2^{-4} = \frac{1}{16} = 0.0625$$

For instance, the binary fractional number $(0.1101)_2$ represents:

$$(0.1101)_2 = (1 \times 2^{-1}) + (1 \times 2^{-2}) + (0 \times 2^{-3}) + (1 \times 2^{-4}) = 0.5 + 0.25 + 0 + 0.0625 = 0.8125_{10}$$

How to Convert Decimal Fractions to Binary

Converting a decimal fraction to binary follows the repeated multiplication-by-2 method:

1. Take the fractional part of your decimal number.
2. Multiply the fractional part by 2.
3. The integer portion of the product (either 0 or 1) becomes the next binary fractional bit.
4. Subtract the integer part to keep only the new fractional remainder.
5. Repeat the multiplication until the remainder reaches 0 or until you reach the desired precision in bits.

Let us convert $0.625_{10}$ into binary:

• $0.625 \times 2 = 1.25$ → Record bit 1, remainder is $0.25$
• $0.25 \times 2 = 0.50$ → Record bit 0, remainder is $0.50$
• $0.50 \times 2 = 1.00$ → Record bit 1, remainder is $0.00$ (Terminated)

Combining the recorded bits gives $(0.101)_2$.

Finite vs Repeating Binary Fractions

In base 10, a fraction has a terminating decimal expansion only if its simplified denominator has prime factors of 2 and 5. In base 2, a fractional number terminates if and only if its denominator is a pure power of 2 ($2^k$).

Fractions like $\frac{1}{10} = 0.1_{10}$ or $\frac{1}{3} \approx 0.3333$ become infinite repeating binary fractions ($0.1_{10} = 0.0001100110011..._2$). This is the mathematical reason behind floating-point precision rounding in computers.

Related Binary Tools

Explore other helpful digital logic and base conversion calculators:

• Binary Calculator: Perform addition, subtraction, multiplication, and division on binary numbers.
• Decimal Binary Converter: Convert integer decimal values to base-2 binary format.
• Bitwise Calculator: Compute AND, OR, XOR, NOT, and bit shifts.