# Proof of $+\infty=-\infty$ (Maybe)

I guess we can agree that $+0 = -0$. Now, after that, I was simply looking at some graphs. The graph of $\tan x$ shows asymptotes at x = $n\pi + \pi/2$. I got to thinking, what if they weren't asymptotes, but actually continuous lines?

If I take $0$ and ($+/-$)$\infty$ as diametrically opposite points of a circle, and sort of roll the $\tan x$ graph into a cylinder with the Y-axis as circumference and x-axis as the length of the cylinder, then $\tan x$ will become continuous.

This seems intuitively valid, but is there a formal proof possible that $+\infty=-\infty$? (Using simple mathematics, if possible. I am still in Grade 11)

• Whether it is even true depends on how you define it ($\infty$ and $-\infty$ that is). Mar 4, 2016 at 14:14
• If they were to be assumed "equal", what would happen if we took the limit $e^x$ as $x$ tend to $+\infty = - \infty$ Mar 4, 2016 at 14:14
• @TobiasKildetoft The meaning behind it, as far as I have learnt, is that $-\infty$ is a infinitely large negative quantity, and $+\infty$ is an infinitely large positive quantity. Mar 4, 2016 at 14:16
• Real numbers are used in different contexts. So -\infty and +\infty are used when we want to talk about the ordering of real numbers. We use one infinity when we want to compactify real numbers (but then of course you lose the ordered structure). Looking at tangents to graphs is somewhat misleading, because it assumes tacitly an ambient space where the graph is embedded. For example you can think of tan as a complex function and then the graph is embedded in a high dimensional space.
– DBS
Mar 4, 2016 at 15:42
• Imagine the real line as a circle of infinite radius. Then $\pm\infty$ coincide. Mar 4, 2016 at 17:27

In general (for grade 11), remember that $\infty$ is not a real number. To say that two elements are equal, they need to be equal in some set. That is, they need first be elements in some set. And $\infty$ is not an element in the set of real numbers.

For example, when we say that a limit (of a function) is (equal to) $\infty$ or $-\infty$, all we are saying is that the values of the function can be made as large (positive or negative) as we would like. So, it can be a bit confusing to talk about a limit being equal to $\infty$ because it gives the impression that $\infty$ is a number.

Does that mean we never ever talk about $\infty$ as a number (or element in a set)? No, for more on this see for example

• That's why I had to argue with my teacher for writing intervals as $[a,\infty]$ Mar 4, 2016 at 14:19
• @SS_C4: So, you would not want to write $[1,\infty]$, you would write $[a,\infty)$. The point, again, is that $\infty$ is not a number. by using $]$ with the $\infty$ it is as if you are saying that the interval contains the number $\infty$. Mar 4, 2016 at 14:20
• I can understand the one point compactification, (it's exactly what I was looking for) but it can be reduced to 2D, right?(Sphere to circle) And is it valid in the set of reals? (As $\infty \not\in R$) Mar 4, 2016 at 14:22
• @SS_C4 you do have the concept of the extended real line. This might be what you are looking for: en.wikipedia.org/wiki/Extended_real_number_line Mar 4, 2016 at 14:25
• And maybe the hyperreals too have "infinite" quantities. But Im not sure if one can talk in this context of a generalized infinite quantity. Mar 4, 2016 at 15:36

When looking at only the real numbers, it makes sense so seperate $\infty$ from $-\infty$.

However, in the complex plane, the Riemann sphere is often used to depict infinity.

Simply put, you have only one infinity and many ways to reach it.