Somewhere down the fanfiction road, I’m tempted to talk Twilight Sparkle into dividing by zero, just to see what happens. In the meantime, Mark Alger, no slouch of a storyteller himself, is playing with the concept, starting, sensibly enough, with Dolly telling him you can’t do that:
Well, properly speaking, you can, but the answer is out of the normal bounds of our concepts of numbers. And, of course, computers lose it when you try to make them calculate it. But, really, it makes logical sense. Zero zeroths is a whole zero, right? I mean, it’s nothing, but it’s ONE nothing. A slippery concept, I’ll admit, but not as weird as n dimensions.
And this also requires admitting that dividing zero by itself to get one is a special case. And what if that means that 0/0=1 is also 0/0=∞? Talk about your special cases. And what does that imply about the question raised in the linked article as to whether infinity actually exists in the real world, or is just a mental construct? See how that blows your dress up.
The reason we have mental constructs in the first place, I suspect, is as placeholders for things we actually haven’t found yet. (Think “Higgs boson”; it explains much, even in its “well, we think we saw one” status.) If you push me, I’d say that infinite anything probably violates at least one law of physics — and that a hundred years from now, those laws will have probably been updated somewhat.
That said, there are transfinite numbers, which I understand barely if at all, and hyperreal numbers, which are at least easy to explain:
The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form
1 + 1 + … + 1.
Such a number is infinite, and its reciprocal is infinitesimal.
I never expect to see a number that is truly infinite, though its reciprocal I see every month on my bank statement: it’s the interest rate they pay me.