an infinitely small gap is not a gap at all

Quote from: zero44 on December 11, 2013, 09:39:03 am

To me, it equals 1.

Simply accept that as long as 41.9999...99958=42, I'm right.

That "notation" is heavily misleading and, somewhat unfortunately, crops up whenever this debate does. What does the use of the ellipsis denote? An infinite string of characters extending to the right. The tricky part here is the word "infinite". An infinite string has very different behavior from a finite one. In particular, the notion of an "end" does not exist. You can't append to the end of an infinite string because there isn't an end to append to. By placing numbers after the ellipsis, you're attempting to denote this. It's not a valid operation and hence what you get is wrong. 41.999...=42 would be the only way to write that statement and thankfully, it's true.

To see why, let's take a quick look at what the reals are:

Reals intuitively consist of all the numbers along a number line. This can be formalized with something called a dedekind cut, which is essentially as follows: For every point X along a line, define sets A and B such that A consists of all the points below X and B consists of all points not in A. We can define the reals as every point that we can do this for. Note that each A has the property that while there is an upper bound on the set (X), there is no greatest member of A. In the discussion 0.9999... = 1, what we're really asking is whether there's a a distinct real number from 1 such that there is an infinitesimal difference between them. In other words, is there a greatest member in set A? As I stated earlier, this is a property of the way the set is defined that there is not. Therefore, if 0.9999... is distinct from 1, it cannot be in A. It must therefore be a member of B, as the smallest member. Unfortunately, this is exactly where 1 is located. Thus 0.999... = 1.

If this confuses you, that's fine. It's simply a more intuitive explanation of the formalization of reals. Note that, unlike what shmibs, says, I am an uninteresting person (err, i mean you can have systems where the equivalence is not true), but it's not possible with the *reals*.


EDIT: Thanks to shmibs for edits and *ahem* punctuation.