Is infinity the reciprocal of zero? Is zero the reciprocal of infinity? It would make sense that they would be--they behave in a similar way (anything multiplied by zero or infinity results in zero or infinity, for example) and you can't have a number infinity close to zero but not zero (as far as I know.) Also, my basic understanding of the Riemann sphere seems to imply that since infinity and zero are opposite poles, they must be reciprocals.

I do know that $\frac{1}{0}$ is technically undefined and infinity can't really be treated like another number, but could they be reciprocals in some situations?

So, is $\frac{1}{\infty}$ some infinitesimal, or is it zero? And does $\frac{1}{0}=\infty$?

I'm sorry if this is a stupid and obvious question, my understanding of mathematics in the realm of infinity is... shaky to say the least.

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    $\begingroup$ Context. In the usual real numbers with the usual multiplication, $\infty$ is not a number and so you don’t operate with it, and $0$ has no reciprocal. There are other systems in which $0$ and $\infty$ other meanings which may make them reciprocals of one another (e.g., in the extended complex plane). It’s not a dumb question, but rather it is a question that requires one to be very precise as to what things mean before it can be answered. $\endgroup$ – Arturo Magidin Feb 25 '19 at 2:57
  • $\begingroup$ no. and............ no. $\endgroup$ – Umberto P. Feb 25 '19 at 2:58
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    $\begingroup$ Some reading. You can also check out the duplicate. In short, infinity is a loose collection of vaguely related concepts over many fields of mathematics. It's not typically considered a number, and when it is, you can expect some of the structure we take for granted to break down. There is no multiplication defined on the entire Riemann sphere. $\endgroup$ – Theo Bendit Feb 25 '19 at 3:03

It depends on the number system you're using.

If you're using the real numbers or the complex numbers, then zero has no reciprocal. In other words, $1/0$ is an undefined expression. Also, in these systems, there's no such number as infinity. In other words, $\infty$ is an undefined symbol.

If you're using the projectively extended real line or the Riemann sphere, then the reciprocal of zero is infinity, and the reciprocal of infinity is zero. In other words, $1/0 = \infty$ and $1 / \infty = 0$. (Note that the reciprocal of infinity is exactly zero, not infinitesimal. None of these four number systems contain any infinitesimal numbers.)

Out of these four number systems, the first two (the real numbers and the complex numbers) are much more commonly used than the last two (the projectively extended real line and the Riemann sphere). So much so, in fact, that we usually say "division by $0$ is undefined" and "infinity is not a number" without clarifying which system we're using.

The reason that the first two systems are more commonly used is that these two systems are fields, and the other two are not.


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