Infinity is certainly a “number”; https://en.wikipedia.org/wiki/Projectively_extended_real_lin... https://en.wikipedia.org/wiki/Riemann_sphere

It's not a number if you are like me and say "a number is an element of a ring." (We're not talking about ordinals or cardinals, the subject being rationals, otherwise I might have a different definition.)

The projectively extended real line doesn't support adding infinity to itself, which is what the grandparent to your comment was talking about. The problem is that the projectively extended real line does not have both a positive and negative infinity.

Polynomials are numbers now? :-)

Mainly, I was poking a bit of fun at the idea that "number" even has a rigorous definition. You can't really assert whether infinity is a number or not without 1) knowing what they mean by "number" or 2) knowing what they mean by "infinity." If you want "numbers" to be a field extension of the reals and you want it to contain an "infinity," then the hyperreal numbers might be appropriate [1]. If you are instead content with having an infinitesimal and having only a ring extension, then the dual numbers might be appropriate [2].

[1] https://en.wikipedia.org/wiki/Hyperreal_number [2] https://en.wikipedia.org/wiki/Dual_number

But, anyway, if you can claim infinity is a number because someone thought of the real projective line, I can say polynomials are numbers. Square matrices, too --- I think of square matrices as being big numbers; rank measures how invertible a big number is.

A stranger consequence of my definition is that a continuous real-valued function on a space X is a "number." If X is a single point, then such a function is the same as a real number. If X is a discrete set of n points, then the set of functions is R^n. If X is compact (I think that's sufficient?) then the maximal ideals in the set of continuous functions is in correspondence with X itself. This suggests that for any ring, one may imagine there to be a space that it is the functions of, whose points are the maximal ideals of that ring. For the ring of complex one-variable polynomials, the points correspond to C itself (the maximal ideals are generated by (x-c) for varying constants c). So, yeah, polynomials are numbers now.

> In real analysis, the projectively extended real line [...] is the extension of the number line by a point denoted ∞.

> The extended complex numbers consist of the complex numbers C together with ∞.

It may be an "extended number", but it is not a number.

If you can consider a “complex number” or a “transcendental number” to be a “number”, then there’s really no reason to not also consider ∞ to be a number.

In general, the boundary of the category of ideas (if you like, elements of some formal model) that can be called “numbers” is a very fuzzy and arbitrary one.

Some people might reject as “numbers” anything other than the counting numbers 1, 2, 3, 4. Others might allow “negative numbers” or ratios. Still others are happy to include quaternions or infinite strings of digits output by some computer program. Meh.