How to define $-\infty$?

Question

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 $-\infty$?

Answer 1

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 $\mathbb{R}$).

Answer 2

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.

Answer 3

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