> After lunch, I went to teach limits to first years students, and with a total straight face I told them that ∞ is not a number.
When you apply Alexandroff extension to add the point at infinity to, say, the real numbers, what you're left with is not a set of numbers (i.e. a field) anymore. So it makes sense to say that ∞ is not a number. Moreover, the way ∞ is used in analysis is different from Alexandroff compactification, in that you usually use two infinities (±∞) as a shorthand for quantification over increasing or decreasing sequences of real numbers (this can be formalized using extended real numbers [0] or other gadgets but doing so has no advantages in a first-year analysis class, and might in fact make matters worse).
It was a long time ago, something like an optional course in Advanced Functional Analysis. It was about the algebras of functions with and without unity, and how to complete the ones without unity using the compactification (i.e. including a ∞) and a few variants.
> two infinities (±∞)
It depends. In the real numbers it depends, but in most cases I agree that it's better to use two. In complex analysis it's much better to have only one infinity. And there are more weird case like the projective plane where you have one infinity in each direction.
> So it makes sense to say that ∞ is not a number.
I agree, it's not longer a field and the operation lose many properties if you try to extend them. So I said "(almost) a number". Anyway, the weird part is that in some cases you can write f(∞) in an advanced math course, but you can never write f(∞) in a fist year math course.
When you apply Alexandroff extension to add the point at infinity to, say, the real numbers, what you're left with is not a set of numbers (i.e. a field) anymore. So it makes sense to say that ∞ is not a number. Moreover, the way ∞ is used in analysis is different from Alexandroff compactification, in that you usually use two infinities (±∞) as a shorthand for quantification over increasing or decreasing sequences of real numbers (this can be formalized using extended real numbers [0] or other gadgets but doing so has no advantages in a first-year analysis class, and might in fact make matters worse).
[0] https://en.wikipedia.org/wiki/Extended_real_number_line