Reviews:
Remember conversion of decimal integers to binary using division method, binary to decimal using multiplication method.
Fraction conversion is the other way around: we start with a decimal fraction, and keep multiplying by two. The result of each multiplication is one bit of the result.
Key point in converting decimal to binary: remember that in converting ints from decimal to binary, we worked by continually picking off the least significant bit by asking if the number is odd, and then dividing the number by two to get the next bit. We'll work here by continually asking if the number is less than 1/2, and grabbing a 0 if it is or a 1 if it isn't.
A good way to organize the problem is to use a table, with columns old, bit, and new. We start with the number we're planning to convert in the top row old entry.
Now, we work across each row, and multiply the old entry by 2 to get a new value. Now we take the integer part of the new value and put it in the bit entry, and put the fraction part in the new entry. If the new entry is non-zero, we copy it into the old entry in the next row down and continue (if it is 0, we're don).
The result is the contents of the bit column, with the most significant bit in the top row.
Here's an example: let's convert .375 from decimal to binary.
| old | bit | New |
|---|---|---|
| .375 | 0 | .75 |
| .75 | 1 | .5 |
| .5 | 1 | 0 |
So the result is .011.
You keep it up until you either have converted the number, or you've generated a bit string that is as long as the space you have available to put the result. To see an example of a number where you will have to quit early, try converting 0.2. The result, in binary, repeats.
Likewise, we can convert binary fractions to decimal by using a division method. We'll build a table like before; this time the columns are old, bit, and new. The top-row old value is 0, and we put one bit in each row of the bits column with the least-significant bit first.
Now, we go across each row, and add the old value to the bit value and divide by two to get the new value. Then we copy the new value to the old value of the next row, and keep it up until we've filled in all the rows. The answer is in the bottom new value.
Here's an example, using .0112.
| old | bit | (old + bit)/2 |
|---|---|---|
| 0 | 1 | .5 |
| 0.5 | 1 | .75 |
| .75 | 0 | .375 |
So the result is .37510.
As with integer conversions, we can do this by converting groups of bits. But, instead of starting from the right, we start from the left. A way to remember this is that in both cases we start grouping from the radix point; another way is that with integers leading zeros are insignificant but with fractions it's the trailing zeros that are insignificant.