Consequences of alternative number system in which $.999\dots \neq 1$ I'm sure many of you have encountered friends or internet strangers who refuse to believe that $.999\dots = 1$. Often I concede that there can, in fact, be number systems in which the two are not equal. BUT the very standard convention is a system in which the two are equal. 
Timothy Gowers writes: 

However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.

Can anyone give examples of such alternative number systems, and what familiar rules "break" in these systems? 
 A: This might be beyond the scope of this question but, define the "line with two origins" as follows. Take the quotient space $\mathbb{R}\times\{a\}$ and $\mathbb{R}\times\{b\}$, with equivalence relation $(x,a)\equiv (x,b)$ iff $x\neq 1$, where $a,b$ are real numbers. This space is effectively equivalent to $\mathbb{R}$, in that there is a single point for each real number $r$ not equal to 1. The difference is that the point $1$ has two copies: $(1,a),(1,b)$. You can easily define addition, subtraction and multiplication in this space, i.e. $(x+y,a)=(x,a)+(y,a)$, $(kx,b)=k(x,b)$, etc. You might have to scratch your head about defining $(x,a)+(y,b)$ .
This space is non-Hausdorff, in that all neighborhoods of $(1,a)$ intersect $(1,b)$ and vice versa. So while the two versions of 1 are different, there is no sensible separation between the two. Now we can define $0.999\ldots$ to equal $(1,a)$, but not $(1,b)$. So now you have two versions of 1 at your disposal. I'm not sure how useful this whole setup is but I hope it proves that it's just plain sensible to define $0.999\ldots=1$ and be done with it. Spaces where limits are either not unique or where you have this adhoc duplicity become a nightmare to work with.
A: Jam has covered in the comments some of the "abandon some of the familiar rules of arithmetic" part of the statement. An example "strange new objects" Gowers is referring to are probably hyperreal numbers.
What are hyperreal numbers? Firstly, all the real numbers are hyperreal numbers, so we might feel we know quite a lot of them already. Let's find some more interesting examples.
We want $0.999...$ to be different from $1$. Question: what, then, is $1-0.999...$? Naively, it looks like $0.000...$, which is $0$ as a real number, so we need to be a bit more precise to characterise the difference.
We can describe $0.999...$ more precisely as the limit of the sequence
$$ (0,0.9,0.99,0.999,\dotsc). $$
Equally, $1$ can be described as the sequence $(1,1,1,1,\dotsc)$, and indeed, any ordinary real number can be defined in this way. So if we assume we can do arithmetic on these sequences in the usual way, $1-0.999...$ is
$$ (1,0.1,0.01,\dotsc). $$
Let's call the number represented by this sequence $\epsilon$.
We can impose a partial order on such sequences by saying that $x>y$ if the terms of the sequence corresponding to $x$ are all eventually bigger than the terms in the same position in the sequence corresponding to $y$. So, for example, $0<0.999...<1$. But we also have $0<0.9<0.99<0.999<\dotsc<0.999...$. So for any integer $n$, $ 1-0.999... = \epsilon < 10^{-n}$. Assuming we can divide,
$$ \frac{1}{\epsilon} > 10^{n}. $$
This can't happen in the reals: the Archimedean property/axiom explicitly prevents this. What we have created is a number larger than any power of $10$, and hence larger than any natural number: in other words, an infinite number. $\epsilon$ is called an infinitesimal.
The hyperreals contain more than one infinite number: indeed, there are (uncountably) infinitely many of them: $2/\epsilon$ is a different infinite number, $1/\epsilon^2$ is bigger than both of them, and so on. We have (clumsily and scratchily, with some rather careful handwaving) cooked up a new number system that contains infinities and infinitesimals, and it is these that are one example of Gowers's "strange new objects". Strange because we can have a consistent arithmetic of infinity, doubly strange because they have even less to do with the real world than the reals!
The primary use of hyperreals is in the field of non-standard analysis, where the hyperreals may defined much more coherently and carefully, normally using a gadget called a non-principal ultrafilter. (Why do we need this? It's an extension of the same problem we have in the reals with the very numbers we have been discussing: defining hyperreals using sequences, we want to know when two sequences define the same number [is $(0,1,1,1,\dotsc)=(1,1,1,\dotsc)$? It probably should be, but we haven't defined equality to allow this.]. Terry Tao has a nice post (coming up on 10 years old now) that explains the rôle of the non-principal ultrafilter in terms of "voting": which elements of the sequence we allow to have a say in what the whole sequence represents.) The jewel of nonstandard analysis is the transfer principle, which allows us to prove results about real numbers by showing they hold for particular sets of hyperreal numbers.
A further example of extra-real numbers is Conway's surreal numbers, which are essentially the largest set of objects you can make with a Dedekind cut-like construction. They too contain infinitesimals and infinite numbers (and are in fact contain the hyperreals, the ordinals and various other extensions).
