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$\infty \notin \Bbb R$ that’s true.

Plus and minus sign are characteristics of real numbers so what is the evidence that infinity has also the same characteristics $+\infty$ and $-\infty$.

We can see that complex numbers have different characteristics, for example, we cannot say there is positive and negative complex number but there is different system such as:

For $C=a+ib$ we can change the plus\minus sign to have four different numbers as, $$a+ib, a-ib, -a+ib, -a-ib$$ Then infinity should be something different, totally incompatible with the meaning of “number”. Therefore, there is no meaning for $+\infty$ & $-\infty$

And if we consider them as two positive and negative infinities we should confess they are still lying on numbers line, otherwise it is meaningless as if someone add a sign to the letter "A" to become "+A" or "-A".

"A" is not a number and there is no meaning to add it a sign.

Personally I see three are different infinities and I define them as below (these definitions are just redefining the infinity to avoid the problem of plus\minus sign):

Positive infinity: it is the biggest number stays at the end of real numbers line and after that point starts the field of infinity.

Negative infinity: the same as Positive infinity but to the left side of negative numbers.

Infinity: a mathematical idea of a field lies outside real numbers line and has no sign.

And that is just individual thinking of definitions for infinity.

Please what is exactly the meaning of signs of $+\infty$ and $-\infty$ if they are not numbers?

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    $\begingroup$ Dogs are not humans. Why does a dog have a heart? $\endgroup$ – Asaf Karagila Nov 20 '16 at 7:00
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    $\begingroup$ Both $-\infty$ and $+\infty$ are symbols, the sign has no "numeric" meaning. $\endgroup$ – user261263 Nov 20 '16 at 7:00
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    $\begingroup$ Formally speaking, so do humans. But you're just reenforcing my point. Two things which are not the same could have a shared property. So while $\infty$ is not a real number, it can has a sign, like a real number. $\endgroup$ – Asaf Karagila Nov 20 '16 at 7:04
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    $\begingroup$ @Pentapolis You should stop seeing $+\infty$ as an entity formed from a sign: $+$ and a symbol: $\infty$. There is, actually, a single atomic symbol: $+\infty$ $\endgroup$ – user261263 Nov 20 '16 at 7:15
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    $\begingroup$ Saying that "having a sign" means that you "lie on the numbers line" is the same as saying that "having a heart" means that you "are human". Which, as we agreed, is not the case. $\endgroup$ – Asaf Karagila Nov 20 '16 at 7:22
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In the simplisitic sense, $\infty$ signifies "greater than any real number". But the real numbers also extend to the negative. So we also want to talk about "less than any real number".

In that case, we use $+\infty$ and $-\infty$ to mean exactly that. Two points which formally mean we've "went to far" in either the $+$ direction or the $-$ direction of the real numbers.

Note, by the way, that having a sign does not mean that you're necessarily a real number. And being a number does not mean that you're automatically going to have a "sign". There are fields, like the hyperreals or surreal numbers which are ordered fields, where a sign makes sense (order-wise, not just from an algebraic point of view as an additive inverse); and there are fields like the complex numbers or finite fields, which cannot be made into ordered fields, and therefore a sign has no actual meaning to it.

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  • $\begingroup$ You are talking abouts limits, and I agree with you if you consider $+\infty$ and $-\infty$ as if we went so far in both directions. $\endgroup$ – Pentapolis Nov 20 '16 at 7:13
  • $\begingroup$ Infinity is a mathematical idea. and the word "Larger" is not good approach. $\endgroup$ – Pentapolis Nov 20 '16 at 7:15
  • $\begingroup$ Larger is a mathematical term. If you have a linear (or even partial) order, then in that context $x$ is larger than $y$ if $y<x$. This might not be a strict definition, but it is a very common term. In the case of the real numbers, there is a natural linear order to consider. $\endgroup$ – Asaf Karagila Nov 20 '16 at 7:17
  • $\begingroup$ I don't like larger either, in some sense $-3$ is larger than $-1$ as its absolute value is 'bigger'. But you couldn't argue that $-3$ was greater than $-1$. So I'd say $\infty$ is greater than any real number. $\endgroup$ – Dan90 Nov 20 '16 at 8:28
  • $\begingroup$ @Dan: In some sense, sure, but that's not the natural interpretation of the word. You cast it into a different term, which admittedly is the problem of informal terms. But greater works great. Any suggestions for the negative one? $\endgroup$ – Asaf Karagila Nov 20 '16 at 8:31
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You don't need to have a number to define + or -, take for example a group $G$, if you denote by + it's associative binary operation, then each element of the group has an inverse, this is if $x \in G$ there exist $-x \in G$ such that $x + (-x) = e$ where $e$ is the neutral element and $x$ is not necesarily a number. Also there are indeed different infinities but not in the way you see it. The smallest infinity is the amount of natural numbers or integer numbers, where as the amount of real numbers is a much bigger infinity. An other way to look at minus infinity is if you have a diverging secuence $\{a_n\}$ which tends to infinity then the secuence $\{-a_n\}$ tends to minus infinity and these two infinties would be equally large. Maybe in a book of elementary set theory ir is better explained.

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Sometimes notation is developed so that we can write things down concisely. Clearly, the asymptotic behaviour of $\frac1{x}$ is different whether you approach $0$ from the left or the right, so we write $\lim_{x \to 0^+} \frac1{x}$ as $\infty$ and $\lim_{x \to 0^-} \frac1{x}$ as $-\infty$. One way to view the answer to your question is as describing these different behaviours.

It could be a matter of convention; some textbooks allow $\lim_{x \to \infty} x = \infty$ but others do not allow this statement of equality (and express it in a different way). Either way, $\infty$ is not a number but serves a useful notational purpose. Unlike the example of A and -A, adding positive or negative here does provide meaningful information about what's going on.

The point is, all notation is essentially invented. We could have written negative numbers as upside down, the notation means what we say it does and it can be extended when necessary, along with a proper definition. Say I form a group from a set of objects (not necessarily numbers) $\{a,b,c\}$ and define all combinations with a weird operation $\circ$. What would $-a$ mean here? Nothing, unless I defined it to be something. Math notation is not rigid, it can be fluidly defined to suit the needs of the mathematics, which may then be adopted widely so that all are working on the 'same page'.

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There is a "theory" where a single $\infty$ (without sign) is used: it is the so-called Alexandrov compactification. This could be called also a 1D stereographic projection, giving this new space the topological properties of a "ring" (i.e., homeomorphic to the unit circle).

In fact, $\mathbb{R}$ can be "compactified" in two ways, by adjoining it a single $\infty$ or a double infinity $-\infty$ and $+\infty$.

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  • $\begingroup$ There is a theory of Riemann sphere en.wikipedia.org/wiki/Riemann_sphere $\endgroup$ – Pentapolis Nov 20 '16 at 8:27
  • $\begingroup$ Yes. What I would like to stress is that it is a matter of topological conventions. For example, it is striking that when you take for example the limits at infinity of an homographic expression $\frac{2x+3}{x-5}$, you find the same limit, 2, either in $-\infty$ or $+\infty$. In such cases, you wonder if a single infinty would be enough... And is a perfectly possible... $\endgroup$ – Jean Marie Nov 20 '16 at 8:35
  • $\begingroup$ @Pentapolis Then there is also the extended Euclidian plane model of the projective plane. But none of that explains why you'd think that the $\pm$ works as a "sign" for infinity in the sense you asked. $\endgroup$ – dxiv Nov 20 '16 at 8:36
  • $\begingroup$ And what I would like to say we should consider $+\infty$ and $-\infty$ as limits so that they are still numbers. $\endgroup$ – Pentapolis Nov 20 '16 at 8:43
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If we consider the following function $y=\frac{x}{1+\mid x \mid}$

Function

Then the limit this function as $x \rightarrow +\infty$ is 1, while the limit as $x \rightarrow -\infty$ is -1.

This is really all we mean when we talk about positive or negative infinity.

You seem to be using the analytical style of a philosopher. That has its place, but I suggest that the best way to get an understanding of mathematical concepts is to see how they function within the context of problems.

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  • $\begingroup$ I can also give you an example of a limit around number one give two different values for $1^+$ and $1^-$, so does it mean that we have two numbers of $1$?, absolutely no. $\endgroup$ – Pentapolis Nov 20 '16 at 8:39
  • $\begingroup$ I agree. This is why I explained in my own answer that, placing it in the "purely mathematical point of view", new spaces are necessary to deal with these issues, either by adjoining one symbol $\infty$ or by adjoining two new symbols $-\infty$ and $+\infty$ to $\mathbb{R}$. $\endgroup$ – Jean Marie Nov 20 '16 at 8:41
  • $\begingroup$ @Pentapolis the implication is not there there are two number 1s, but that there a two different limiting behaviours around a single point. By 'different' that means we have two different objects - the limits could be 4 and -10. Why can't the different limits be $+\infty$ and $-\infty$? $\endgroup$ – Dan90 Nov 20 '16 at 8:49
  • $\begingroup$ Two points: If we are describing and interval, then $(-\infty,5]$ describes the set of numbers that is unbounded on the low side and goes up to and including 5. It makes more sense to say that than to write $(\infty,5]$ Second, you could take the concept further to complex infinities. The limit as $x \rightarrow i \infty$ of the function above is i. $\endgroup$ – David Elm Nov 20 '16 at 8:57

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