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Why don't we include $\pm\infty$ in $\mathbb R$?

If we do so, many equations will got real solution (e.g. $2^x=0$), and $\mathbb R$ will be much more complete. Why don't we do so?

Thank you.

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    $\begingroup$ $\mathbb R$ is a field. The extended real number system $\overline{\mathbb R}$ is not a field. Both systems have their uses. $\endgroup$ – GEdgar Jun 9 '13 at 12:34
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    $\begingroup$ Because algebra starts failing. What is $\infty - \infty$ or $0 \cdot \infty$? $\endgroup$ – Najib Idrissi Jun 9 '13 at 12:34
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    $\begingroup$ @user42912: We can define something, yes, but there no nice and satisfying definition. For example, what would you define $\infty - \infty$ as? $\endgroup$ – Najib Idrissi Jun 9 '13 at 12:38
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    $\begingroup$ @user42912: But then $\infty - (\infty + 1) = \infty - \infty = 0 \neq (\infty - \infty) + 1 = 1$. So you lose associativity. There are many problems like that with infinity. $\endgroup$ – Najib Idrissi Jun 9 '13 at 12:47
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    $\begingroup$ @BandeiraGustavo It means if you have an expression like $a + (b + c)$, then you cannot algebraically rearrange it to $(a + b) + c$, which could be prohibitively limiting. $\endgroup$ – Peter Olson Jun 9 '13 at 20:14
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A large reason why we don't include $\infty$ is because we can't really do arithmetic with it. $\mathbb{R}$ is a field, meaning that is satisfies a list of axioms that give it a certain structure. I suggest you look up these axioms if you're not familiar with them and see just how many of them start to fail if you try to include an $\infty$ in $\mathbb{R}$.

Yes, including $\infty$ may give you a few extra solutions to some problems, but it won't solve every problem ($x^2=-2$ for example) and in some cases, it will introduce answers you probably don't want (would $x=x+2$ now have the solution $x=\infty$?). All of this isn't really worth all of the issues you get for including $\infty$ in $\mathbb{R}$.

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By defining, for example, $+ \infty + n:= + \infty\ \forall n \in\mathbb{R}$ , we'd lose the group structure and this is no good for lots of purposes.

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There is no reason we can't add $\pm\infty$ to our set. Sometimes it is actually convenient to do so: in measure theory, it's common to work with "the extended non-negative reals" $\mathbb R_{\geq 0}\cup\{+\infty\}$ because it simplifies some common statements.

For doing arithmetic, it is very inconvenient however. In our standard $\mathbb R$, whenever we divide by something, we always have to make sure the denominator is not $0$, since dividing by $0$ is not allowed. This is an annoying artifact we have to live with as long as we want to divide things. Likewise, if we added $\pm\infty$ to our set, there is no good definition for what $\infty-\infty$ is, so whenever we subtract two variables, we have to make sure our numbers are not $\infty$. If we add two numbers $x+y$, and we happen to have $x=\infty$ and $y=-\infty$, we run into the same problem, so we can't even add two numbers without first checking they're not infinite! This is tedious, so we just decide not to include $\pm\infty$ in our set.

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While it is true that we could have "real" solutions to equations such as $2^x = 0$ in $\Bbb R \cup \{\infty\}$, we can get a similar amount of intuition just by saying that

$$ 2^x \rightarrow 0 \;\operatorname{as}\; x \rightarrow \infty $$

while not loosing a fair few important tools in analysis (well, maybe not in non-standard analysis, but I have little working knowledge about it) - how can you use the so-called "Axiom of Archimedes", which states that $\Bbb N$ is not bounded above in $\Bbb R$? While you could adapt this so it only holds for the non-infinite parts of our new number system, then what is the point of it? However, without such a tool, I don't see how it is possible to prove much in the system; I'm fairly sure something like the Bolzano-Weierstrass theorem will not hold in $\Bbb R \cup \{\infty\}$.

Also, the comments of nik are also worth thinking about - is it possible to define $\infty$ in a way which doesn't lose the associative property of the ordinary reals?

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We do have several extensions to the "regular" real numbers including some sort of infinity.

I think of real numbers as just another step in a iterative construction, and thus there is no need to stop there.

We may start with the natural numbers: $1,2,3,...$ so we can do addition ,e.g. $1+2=3$, and multiplication, e.g. $2\times3=6$, but, say, we can not do every subtraction we may wish: $3-2=1$ is fine, but $2-3=??$ is not.

We extend it to integer numbers: $...,-3,-2,-1,0,1,2,3,...$ and now subtraction is OK for every element: $2-3=-1$

And extend to rational numbers to allow for division of integers (with the exception of dividing by 0). Somewhat surprisingly this is not the whole of real the numbers, so we drop in the oddballs ($\sqrt{2}$, $\pi$, etc.) and we get all the real numbers.

There is no reason to stop there, and we don't. There are quite a few generalizations and extensions. The extended real number line may be similar to what you seem to be looking for. The surreal numbers are also quite interesting, I think - and I hope I can get to read more about them someday as they somehow fit the way I replied myself, years ago, the same question you're asking now.

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There is a lovely extension to $\mathbb{R}$ sometimes called max-plus (see https://en.wikipedia.org/wiki/Max-plus_algebra) in which one adds in $-\infty$ and redefines the key operations to get a very nice semi-ring. The great thing is that this system is very useful in solving a variety of problems in scheduling theory, and discrete dynamical systems. It allows one to solve problems such as deciding the maximum time of completion of a manufacturing process. (The sums are fun and make one think about the algebra used in solving equations in ordinary maths as well.)

If you prefer to add in $\infty$ you can consider (min-plus) and sometimes yoou need to complete in both directions! (see the links at https://en.wikipedia.org/wiki/Max-plus_algebra for lots more.)

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