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It occurred to me that there are some situations (e.g. division by 0 and the tan function) where the two limits of a value, approached from below and above, are negative infinity and positive infinity. And that perhaps instead of viewing negative infinity and positive infinity as two different values/numbers, with the limit not existing in these situations, we could see them as the same number at one end of a circle with zero on one side and positive/negative infinity on the opposite side. I'm curious as to the utility or validity of this approach, and if it has any precedent.

Mark S.
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  • Distinguishing between poles with sign changes and poles without sign changes is important for the discussion of a function. So, I cannot see any merit to consider $-\infty$ and $\infty$ to be the same. Perhaps you mean the extended real line. No, it does not make division by $0$ meaningful. – Peter Aug 08 '21 at 07:39
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    $\mathbb{R}P^1\cong S^1\cong\mathbb{R}\cup{\infty}$ is meaningful geometrically, as is $\mathbb{C}P^1\cong S^2\cong\mathbb{C}\cup{\infty}$. – user10354138 Aug 08 '21 at 10:53
  • @Peter, a nonzero divided by $0$ is often taken to be the single unsigned infinity when you have one, as in the projective real line or the Riemann sphere. The OP is right that 1/x extends to a perfectly good/continuous function from the projective line to itself, sending $0$ to $\infty$ and vice versa. – Mark S. Aug 08 '21 at 12:53
  • See @user10354138's answer. For more information, look up "projective line". I think user should have written the word "projective" in there, and not just symbols that the OP obviously does not know. – GEdgar Aug 08 '21 at 13:07

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Yes, this has been considered; the most common name for it which captures its properties and will help you look up more information is the "projective real line" (as opposed to the extended real line with both $\infty$ and $-\infty$).

But if you don't care about geometry or algebra and just the vague effect this has on shapes, then topologically, this would be the Alexandroff/one-point compactification of the real line (as opposed to the Freudenthal/end compactification).

Mark S.
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