Decimal fraction to binary fraction conversion

1. Successive multiplication method is used to convert a given fractional decimal number to its equivalent binary fraction.
2. In this method of conversion the fractional part of the given decimal number is multiplied by 2 (i.e. the radix of binary number system).
3. The product obtained has an integer part and fractional part, the integer part here is also referred as carry.
4. The carry that we obtain at each multiplication iteration becomes a digit in the fractional binary number.
5. The fractional part obtained is again multiplied and the process is repeated until the fractional part becomes zero or the number of multiplication iteration equals the number of significant digits after the decimal point in the given fractional decimal number.
6. The carry that we obtain at each stage are taken from first iteration to the last iteration to form the numerals in the fractional binary number i.e. the carry obtained in the first multiplication iteration is the most significant bit (MSB) after the decimal point and the carry obtained in the last multiplication iteration is the least significant bit (LSB)
7. This procedure is illustrated in the following example.

Ex1: Convert (0.625)₁₀ decimal number to binary number (?)₂ using successive multiplication method

[0.625 as a fraction in binary from decimal]

1st Multiplication Iteration

Multiply 0.625 by 2

0.625 x 2 = 1.25(Product)         Fractional part=0.25            Carry=1    (MSB)

2nd Multiplication Iteration

Multiply 0.25 by 2

     0.25 x 2 = 0.50(Product)         Fractional part = 0.50            Carry = 0   

3rd Multiplication Iteration

Multiply 0.50 by 2

      0.50 x 2 = 1.00(Product)         Fractional part = 1.00            Carry = 1  (LSB)   
   

The fractional part in the 3rd iteration becomes zero and hence we stop the multiplication iteration.

Carry from the 1st multiplication iteration becomes MSB and  carry from 3rd iteration becomes LSB.

Hence, the fractional binary number of the given fractional decimal number (0.625)₁₀ is (0.101)₂.