# How to define $-\infty$?

I think I understand the fundamental concept of infinity. Elementary mathematics define $\infty := \frac{x}{0}$, for every $x$. And also $\infty := \frac{-x}{0}$ for every $x$. I know only one definition of $-\infty$ as $-\infty= 0-(\infty)$. Is there any other way to define $-\infty$?

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I sure hope elementary mathematics does not define infinity this way. –  Srivatsan Sep 14 '11 at 11:49
To expand on Srivatsan's comment: Infinity is not one well-defined thing. In various areas of non-elementary mathematics, you can speak about abstractions that can be interpreted intuitively as "there's nothing finite to put here, but something different with such-and-such properties". But there are many different variants on this, and none of them claim to be the infinity (as there's no such thing). Some of these concepts can be notated with "$\infty$" by convention within the field they are used in, but that's just convenient notation with limited applicability. –  Henning Makholm Sep 14 '11 at 11:56
@Srivatsan +1. I agree that infinity is not defined as such a way. I just put a commonly used formulation. –  gaurav Sep 14 '11 at 12:02
Srivatsan, I sure hope infinity can be defined in an elementary way. –  Dan Brumleve Sep 14 '11 at 12:06
@Dan, you can probably define an infinity in an elementary way, as long as you don't think your definition captures everything everyone wants to say about things that are not finite. –  Henning Makholm Sep 14 '11 at 12:15

Infinity is not defined in the way you described; something similar can be defined with limits but I think it is a confusing approach.

Here's a more formal definition: $\infty$ and $-\infty$ are points added to $\mathbb{R}$ in such a way the for all $a\in\mathbb{R}$ we have $-\infty < a < \infty$. Topologically speaking, open balls around $\infty$ are subsets of the form $\{x\in\mathbb{R}|x>a\}$ for a given $a$, and open balls around $-\infty$ are subsets of the form $\{x\in\mathbb{R}|x<a\}$.

This allows to formally define concepts like tending to $\infty$ or $-\infty$ with the usual topological approach, and the extanded $\mathbb{R}$ is still a linearly ordered set (although it is no longer a field since arithmetic involving $\infty$ will no longer preserve the nice properties it has in $\mathbb{R}$).

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$-\infty$ can be defined as the surreal number $\{\emptyset|-\mathbb{N}\}$.

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Thanks. It's new to me. –  gaurav Sep 14 '11 at 12:10
Well that is in fact $-\omega$. Stricly speaking, $1/0$ has no solution even in the surreal numbers, nor in any other field. If we extend this to a set model with proper classes, then $-\infty = \{\emptyset|\mathbf{No}\}$ –  rewritten Oct 9 '12 at 9:07

In the real line you can "define" $\infty = \frac{1}{0^+}$ and $-\infty = \frac{1}{0^-}$, but these are really limits: $$-\infty = \lim_{x\to0^-} \frac 1x$$ Here $x\to0^-$ means that $x$ approaches $0$ from the left, i.e., using negative numbers.

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I missed those negative and positive zeros. :) –  gaurav Sep 14 '11 at 12:11