Why can't decimal numbers be represented exactly in binary?

Question

There have been several questions posted to SO about floating-point representation. For example, the decimal number 0.1 doesn\'t have an exact binary representation, so it\'

Answer 1

The root (mathematical) reason is that when you are dealing with integers, they are countably infinite.

Which means, even though there are an infinite amount of them, we could "count out" all of the items in the sequence, without skipping any. That means if we want to get the item in the 610000000000000th position in the list, we can figure it out via a formula.

However, real numbers are uncountably infinite. You can't say "give me the real number at position 610000000000000" and get back an answer. The reason is because, even between 0 and 1, there are an infinite number of values, when you are considering floating-point values. The same holds true for any two floating point numbers.

More info:

http://en.wikipedia.org/wiki/Countable_set

http://en.wikipedia.org/wiki/Uncountable_set

Update: My apologies, I appear to have misinterpreted the question. My response is about why we cannot represent every real value, I hadn't realized that floating point was automatically classified as rational.