converting decimal fraction to binary

[PG1995]

Hi

I'm learning to convert decimal fractions into their binary equivalent.

1: After the decimal point 2's powers are in negative: 2^-1, 2^-2, 2^-3, ...; and this is equivalent to: 0.5, 0.25, 0.125, ....

Let's say we want to convert 0.625 into its a binary fraction.

If we subtract "0.5" or "2^-1" from 0.625 we are left with 0.125 and "0.125" can be represented by 2^-3.

Therefore,
0.625 = 0.5 + 0 + 0.125 = 0.101

The above example was found in a book. Suppose, we have 0.620 instead. How would we convert it into a binary fraction using the method we used above? How do I use this method so that I can deal with any given decimal number?

2: I know there is another method of repeated multiplication by 2. Though this method seems simple but it's not that much natural. Please have a look on the linked scan (or, you can check the attachment):
https://img809.imageshack.us/img809/4655/imgus.jpg

I have tried to apply repeated multiplication method to convert a decimal fraction to a binary one. In case of 0.3125 I was successful but then I tried to apply it on 0.620 without any success. It seems the method would never end.

Please help me.

Regards
PG

 

[PG1995]

It seems now I understand how the method of repeated multiplication by 2 really works. Please check the attachment.

To someone like me who stumbles onto this thread:
The bases which have direct relationship with 2 such as 8, 16, 32, etc. can easily be converted between each other. For example, an octal number such as (562) can easily be converted into a binary one.

562 = 5 6 2 = (101) (110) (010) = 101110010

To see how this works please check the attachment.

PS: Made correction after seeing CZ's posting below. Thanks for pointing this out.

 

[dougy83]

If I wanted to convert a decimal fraction D into a binary fraction B, having N bits to the right of the decimal point, I'd simply multiply D x 2^N. This is the same as the multiply-by-2 method, just done in 1 step (and for a fixed bit size).

So, for 8 bits to the right of the decimal point, 0.625 x 2^8 = 160 = 0b10100000 (or binary 0.10100000 = 0.101). When converting 0.62, it might be the case, as you have found, that it either has a very long or never ending fractional sequence.

 

[carbonzit]

For example, an octal number such as (568) can easily be converted into a binary one.

You do realize that 568 cannot be a valid octal number, don't you?