Is π really a number or is it a computation? For example, fibbonaci(∞) is not a number, and π looks to be conceptually similar. Unlike fibonacci(∞), π has a limit, and we can approximate it with better and better precision, but in both cases the computation will never terminate
If you believe that real numbers are numbers, then, yes, pi is a number. Indeed because pi is computable, it’s actually "more" real than almost all real numbers because there is only a countable infinity of computable reals.
Anyway, in modern math what a real number is, is defined as the limit of a "process", namely a Cauchy sequence. Of course, for the rational subset of reals the limit is trivial.
There are mathematical definitions of the terms "real number", "rational number", etc., but there is no mathematical definition of the word "number".
> we can approximate it with better and better precision
In one of the three common formal definitions of the real numbers, that's what a real number is: a Cauchy sequence of rational numbers, which approximate that real number with increasing precision. (Well, a real is an equivalence class of such sequences.)
(The other two common definitions are the Dedekind reals and the reals as the unique complete ordered field.)
Is 2 a number?
Is 4 a number?
Is 4/2 a number?
Is 3 a number?
Is 3/2 a number?
etc...
All of these symbols represent precise points on the numberline. Pi also represents a precise point on the numberline, so is it not a number?
Ironically, that response runs into the standard problem that many "limit" arguments have.
Generally speaking just because something looks like it's converging from some angle, it doesn't mean that it actually has a well-defined limit, and if it does then it also does not mean that the limit shares the properties of the items in the sequence of which it is the limit.
Examples: 1/n is strictly positive for all n. Its limit for n going to infinity, while well-defined, is not strictly positive. Another example: You can define pi as the limit of a sequence of rational numbers. But it's not rational itself.
So, no, your argument does not prove that pi is a number.
(I'm not arguing that pi is not a number. It definitely is. It's just that the argument is a different one.)
To answer your question you need to define what a number is to you. There are many different kinds of numbers, naturals, integers, rationals, irrationals, computable reals, reals, infinitesimals... Not even getting into complex, quaternions, octonions etc.
Is sqrt(2) a number to you ?
If you accept computable reals as numbers then \pi is definitely a number. So is the golden ratio.