How to Convert Base 10 Fractions in to Binary Decimals

To convert integers from base 10 (decimal) representation to base 2 (binary) representation, one must break up the number into pieces that are powers of 2.

Similar processes can be used to convert non-integers into binary form. This article will show 3 (or 11, in binary) methods of converting fractions into binary decimals.

Difficulty: Moderately Easy

Instructions

1. *Method 1*
   
   To convert a fraction, first convert the numerator and denominator into binary, and then apply binary long division. This is the normal method for converting fractions into base 10 decimals. For example, let´s use this technique to convert 13/22.
   
   First, 13/22 becomes 1101/10110 in base 2. Then, 1101 divided by 10110 is
   0.1001011101000…, with repetition of the digit sequence 1011101000.
   
   This can be cumbersome if you don't like doing long division in base 2, so you may want to try the next method.

2. *Method 10*
   
   Break up the fraction into powers of 1/2, and then use the following list to build the binary decimal:
   1/2 = 0.1
   1/4 = 0.01
   1/8 = 0.001
   1/16 = 0.0001, etc
   
   For example, 13/22 is 1/2 + 1/16 + 1/64 + 1/128 + …, and so
   13/22 = 0.1 + 0.0001 + 0.000001 + 0.0000001 +…
   = 0.100101…
   
   If you find this too time consuming, the next method is the fastest.

3. *Method 11*
   
   Use the binary conversion feature on the computer´s calculator pad or Excel. If you have an older version that does not handle binary decimals, you can get around this with an easy cheat: add 8 zeroes to the end of the numerator, divide, and then move the decimal point 8 units to the left. 8 is just a convenient number, any other number will work as well.
   
   For example, if your calculator won´t divide 1101 by 10110 in binary, then try dividing 110100000000 by 10110 instead. Then,
   110100000000/10110 = 10010111.
   And when you move the decimal 8 units to the left, you get
   1101/10110 = 0.10010111 (truncated)