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

Question

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)

Answer 1

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

• What is the result of infinity minus infinity?
• https://math.stackexchange.com/a/1140678/26188
• https://en.wikipedia.org/wiki/Alexandroff_extension#The_Alexandroff_extension.

Answer 2

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.