Let's take the equation $2x = 1 + x$. We know that in the real numbers we can cancel out an $x$ on both sides (since the inverse of $x$ exists) and we get that $x=1$.
However, if we include infinity (take countable infinity as an example), then infinity would also be a valid solution to the equation. It would also be a valid solution for many other problems, such as $x^2 = x$.
Unfortunately, of course, allowing infinity into arithmetic would cause many "laws", such as $x - x = 0$ for all $x$, to break down. My question is - are there any situations where it might be advantageous to treat infinity as a number?
In measure theory you work with the extended real numbers, that is $\mathbb{R}\cup\{-\infty,\infty\}$, and define $\displaystyle\frac{1}{\infty}=0$, $\infty\cdot\infty=\infty$, $\infty+\infty=\infty$, and a couple of other rules that are consistent with the rest of the normal arithmetic. This is done to avoid having to do special cases when proving theorems. You can read more about it here: http://en.wikipedia.org/wiki/Extended_real_number_line#Measure_and_integration .