[steerablesafe]

I'm well aware of nonstandard analysis, but ∞ is still not a number there, even though there are infinitely many infinitely large elements .

[Spivak]

The extended complex plane is a space where inf is actually number and where division by zero is allowed.

Same with the extended real line.

Infinity is as much a number as it is useful to define it as such.

[steerablesafe]

Ah right, I forget that extending with a single infinity element is useful with complex numbers and with geometry. It's still not very common with the reals alone as +∞ and -∞ are reasonable to want as separate elements there, but it doesn't play nicely with 1/0 that way.