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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?

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There is no problem treating $\infty$ like a number, but if we want to have a group like $(\mathbb{Z})\cup \{\infty\}$ with the addition one would expect it will be very difficult –  Dominic Michaelis Aug 21 '13 at 6:43
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What would happen if you would search the site for this question before asking it yourself? –  Asaf Karagila Aug 21 '13 at 6:46
    
@Asaf I did take a look, but I saw a lot of questions about types of infinity and indeterminate forms. I'm just looking for situations where it might be useful to treat infinity as though it was a number, since I haven't seen any examples of that yet in university. I tried looking up surreal numbers, but they didn't make much sense to me. –  Sp3000 Aug 21 '13 at 6:51
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There are people who attempt to develop non-standard analysis. I have not studied it in detail, but it seems to me that its only real use is for teaching ("concealing" actual details about limits). –  Tunococ Aug 21 '13 at 6:53
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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 .

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where do you need $\frac{1}{\infty}=0$ in measure theory? –  Dominic Michaelis Aug 21 '13 at 6:52
    
Hm, I'm not sure, but I remember seeing it defined. Maybe if you want to allow for extended valued functions it may be useful. –  JLA Aug 21 '13 at 6:58
    
Measure theory, eh. I'll look forward to that in the coming years. I'm seeing that functions like $\sin{x}$ and $\cos{x}$ break down in the extended real number line - I'm guessing that that's related to the fact that they don't converge as $x$ tends to infinity. I guess I can't expect everything to work here. –  Sp3000 Aug 21 '13 at 7:02
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