How to Convert Fractional Decimal to Binary

Converting a fractional decimal number to its binary equivalent is a straightforward process that uses repeated multiplication by 2. This guide provides a clear, step-by-step walkthrough of the conversion algorithm, demonstrates the process using the example decimal number 0.625, and explains how to verify the final binary result.

The Repeated Multiplication Algorithm

To convert a decimal fraction to binary:

  1. Multiply the fractional decimal number by 2.
  2. Record the integer part (the digit before the decimal point, which will always be 0 or 1) as the next binary digit.
  3. Remove the integer part, keeping only the remaining fractional part.
  4. Repeat the process with the new fractional part until it reaches 0.0 or until you reach the desired degree of precision.
  5. Write the recorded integers in the order they were calculated, placed immediately after a binary point (0.).

Step-by-Step Example: Converting 0.625 to Binary

Step 1: Multiply 0.625 by 2
\[0.625 \times 2 = 1.250\]
* Integer part: 1 * Remaining fraction: 0.25

Step 2: Multiply 0.25 by 2
\[0.25 \times 2 = 0.50\]
* Integer part: 0 * Remaining fraction: 0.5

Step 3: Multiply 0.5 by 2
\[0.5 \times 2 = 1.0\]
* Integer part: 1 * Remaining fraction: 0.0 (The process stops here since the fraction is zero.)

Result

Read the recorded integer parts from top to bottom: 1, 0, 1.

Place them after the radix point:
\(0.625_{10} = 0.101_2\)

Verification

To verify the result, convert the binary fraction back into decimal using negative powers of 2:

\[0.101_2 = (1 \times 2^{-1}) + (0 \times 2^{-2}) + (1 \times 2^{-3})\]
\[0.101_2 = (1 \times 0.5) + (0 \times 0.25) + (1 \times 0.125)\]
\[0.101_2 = 0.5 + 0 + 0.125 = 0.625_{10}\]