It is true that different infinities exists[0][1] and there are whole areas of logic examining them.

It is also true that _number_ without any extra qualifier generally means the Real numbers (R) or Complex numbers (C) and those domain do not define infinity as a number, but even then there are only a few good ways to add infinity into each number system:

In R generally you either add a projective point of infinity ∞ [2] that makes geometry sometimes nices or two signed infinities (-∞ and +∞) that make calculus nicer (especially limits and integration)

In C it is often simpler as typically you want to treat them as a sphere and so add an extra point so that the inversion f(x) = 1/x is a well-behaved function. In this domain you often end up working with holomorphic functions[3] and then there is not really an intrinsic difference between a function like f(x) = 1/x and g(x) = x they simply have both a _pole_ f at 0 and g at infinity.

If you want to get trippy even integers can have unusual definitions [4] and then there is always one of my favorite topic in math: surreal numbers [5] (for which I recommend both [6] and [7]) a field where √∞ < ∞/2 < ∞ - 1 < ∞ < ∞ + 1 and is perfectly well defined (but still 0/0 doesn't have any meaning in any of these theories, that is a though nut to crack)

[0]https://en.wikipedia.org/wiki/Ordinal_number [1]https://en.wikipedia.org/wiki/Cardinal_number [2]https://en.wikipedia.org/wiki/Projective_geometry [3]https://en.wikipedia.org/wiki/Holomorphic_function [4]https://en.wikipedia.org/wiki/Algebraic_integer [5]https://en.wikipedia.org/wiki/Surreal_number [6]https://www-cs-faculty.stanford.edu/~knuth/sn.html [7]https://books.google.it/books?id=tXiVo8qA5PQC&redir_esc=y