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Division by zero is not allowed, even in the limit. We can't just say $\lim_{x \to 0}\frac{1}{x} = \infty$. The limit doesn't exist because approaching from the positive side and negative side yield results as different as you can get.

But, there is a circular number system where this does become possible. You then require the number line to be the limit of a really large circle instead of an infinitely extending straight line. Hence, $\infty$ and $-\infty$ become the same and the problem of the limit not existing goes away.

First, what is this circular number system called? I remember reading about it a while back, but can't find it anywhere now.

Second, being able to divide by zero is a nice ability. So, why don't we switch to this circular number system? What are the cons?

Bill Dubuque
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Rohit Pandey
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    Topologically, this is the one point compactification of the reals. – Alan Jul 19 '21 at 06:36
  • What problem do we run into in trying to make it behave like numbers? – Rohit Pandey Jul 19 '21 at 06:39
  • Are you looking for Riemann sphere https://math.stackexchange.com/questions/1403128/the-riemann-sphere-interpretation? – AKP2002 Jul 19 '21 at 06:42
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    Just one obvious example: the $<$ relation no longer works. – MJD Jul 19 '21 at 06:43
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    A less obvious example: there's no $x$ for which $\infty+x=0$, which means that from $a+c=b+c$ you can no longer infer $a=b$. (You might have $1+\infty=2+\infty$.) And to even write “$-c$” or “$x-c$” you would have to show first that $c≠∞$, just as in regular algebra you may not discuss $c^{-1}$ or $\frac xc$ without first showing $c≠0$. – MJD Jul 19 '21 at 19:42

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As a geometric or topological object, this is called the real projective line. But it's not a "number system" because there is no way to make it behave like numbers in any useful way. See Why not to extend the set of natural numbers to make it closed under division by zero? for details of what goes wrong.

MJD
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