Binary Fraction Converter

Converting a fraction between decimal and binary involves two distinct operations: handling the whole number part and handling the fractional part. For the whole number, you repeatedly divide by 2 and record the remainders. For the fractional component, you repeatedly multiply by 2 and extract the integer parts. Mathematically, a binary fraction like $0.101_2$ is evaluated using negative powers of two: $(1 imes 2^{-1}) + (0 imes 2^{-2}) + (1 imes 2^{-3}) = 0.5 + 0 + 0.125 = 0.625_{10}$. The precision of your result depends on the specified number of digits or bit representation limits.

Let us walk through converting the decimal fraction 0.625 into its binary equivalent using standard multiplication by 2. Step 1: Multiply 0.625 by 2, which gives 1.25. The integer part is 1. We keep the fractional part, 0.25. Step 2: Multiply 0.25 by 2, yielding 0.5. The integer part is 0. We keep 0.5. Step 3: Multiply 0.5 by 2, resulting in 1.0. The integer part is 1. Since our fractional part is now 0, the process terminates. Reading our recorded integer parts from top to bottom, 0.625 in decimal becomes 0.101 in binary.

Always watch out for repeating binary fractions; unlike terminating decimals, many simple decimal fractions (like 0.1) result in infinite repeating sequences in binary. Set your bit representation limits ahead of time to handle rounding appropriately. Additionally, always verify that your input string contains only valid binary digits (0 and 1) when converting from binary to decimal to avoid parsing errors.