>>9531728Wow! This thread is horrible. Let's talk about hyperreal numbers.
A hyperreal number is an element of an ultrapower of the real numbers. That is, we take all the sequences of real numbers, and we say that two sequences are equal if they differ at no more than finitely many terms. For example,
{0, 1, 1, 1, 1, ...} = {1, 1, 1, 1, 1,...} since they differ only in the first place.
We embed the reals into the hyperreals by identifying the real number r with the equivalence class of sequences represented by the constant sequence {r, r, r, ...}. We then obtain a natural definition for addition and multiplication with the corresponding identities: pretty much everything is done component-wise!
Now, how do we compare hyperreals to each other? We can't do it componentwise since, for example, {0, 2, 0, 2, ...} and {2, 0, 2, 0, ...} have components which are not consistently bigger or smaller than each other. Basically, we pick a predetermined set of indices on which to compare them component-wise.
Anyway, let's just take for granted that we have a nice (totally) ordered field! Since the sequence {1, 1/2, 1/3, 1/4, ...} is less than {c, c, c, c, ...} on all but finitely many elements regardless of c (as long as it's positive), we can see that {1, 1/2, 1/3, 1/4, ...} is smaller than c but bigger than 0! We've constructed an infinitesimal.
We can construct another just as easily: for example, {1, 1/4, 1/9, 1/16, ...} is clearly between those two. Now we can talk about real numbers and "halos" of infinitesimals around them, and doing analysis on this field and all kinds of fun stuff. However, let's not get distracted.
You ask if 0.999... = 1. In the reals, we define 0.999... = lim n -> infty sum from 1 to n of 9(.1)^n, which by the definition of limit is precisely 1. If that definition holds in the hyperreals, clearly the statement is true as well! But, unlike in the reals, there are numbers infinitesimally close to, but not equal to, 1.