The extended reals are to the reals as [0, 1] is to (0, 1) -- the extended reals complete the real numbers as a topological space by adding two "endpoints" (+∞ and -∞). This topological space cleans up a great deal of calculus. For example, you no longer need to have a separate definition for when a limit diverges to +∞.
However, the key point to note is that the extended reals aren't an arithmetic structure -- we can extend the functions +, -, *, and / to take (some) infinite values, but this extension is via continuity as opposed to any arithmetic meaning.
Another example of an extension via continuity is extending the function (x-1)/(1-x) to equal -1 at x=1, or extending the function (sin x)/x to equal 1 at x=0.
These extensions of +, -, *, and / now need to be thought of merely as functions -- it is generally wrong to try and treat them as arithmetic operations when they take on infinite values.
As an example, the function f(x) = 2x - x cannot have +∞ in its domain, because if we try to evaluate, we run into (+∞) - (+∞), which is undefined.
Why is it undefined? Because no matter what value we assign to it, it would render the - operation discontinuous there. For example, consider:
<br />
(+\infty) - (+\infty) = (\lim_{x \rightarrow \infty} x+1) - (\lim_{x \rightarrow \infty} x) = \lim_{x \rightarrow \infty} ((x + 1) - x) = 1<br />
<br />
(+\infty) - (+\infty) = (\lim_{x \rightarrow \infty} x) - (\lim_{x \rightarrow \infty} x+1) = \lim_{x \rightarrow \infty} (x - (x + 1)) = -1<br />
If - was continuous at (+∞, +∞), then both of these statements would be true -- the first equality by definition of continuity, and the rest as properties of limits.
(This is very closely related to the concept of an "indeterminate form")
