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∞ is not used as a number by mathematicians. Maybe by engineers.

That is not entirely correct. Schmieden and Laugwitz for example developed in the 1950s a nonstandard Analysis which adjoins an infinitely large element (called Ω) to the natural numbers. The basic idea was a formula A(Ω) was true if A(n) was true for almost all finite natural n.

While it wasn't immensely useful going forward, it helped to clarify the use of infinity and infinitesimals in earlier work.

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

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.

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.

Ehh.

https://en.wikipedia.org/wiki/Extended_real_number_line

NB this is hardly nonstandard analysis.

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