A polynomial can be considered a number? According to wikipedia the Euler's number is:
$$e = 1 + \frac{1}{1} + \frac{1}{1\times 2} + \frac{1}{1\times 2\times 3} + \frac{1}{1\times 2\times 3\times 4}+\cdots$$
And I see it's structure is quite similar to the structure of a polynomial:
$$a_nt^n+a_{n-1}t^{n-1}+\cdots+a_1t+a_0$$
Can we consider polynomials as numbers? At least in some specific sense?
 A: Polynomials and numbers are distinct objects. However, a polynomial may be evaluated at a point to give rise to a number. For example, consider the polynomial $x^2-x$. It has value $0$ at $x=1$. Constant polynomials may be identified with numbers though, but they are still distinct in a sense.
A: When you start to consider more general ideas than measuring geometric shapes and counting elements of sets, you start using more general sorts of objects to quantify those ideas. Or sometimes you consider interesting algebraic structures the are analogous to familiar ones, but with some other sort of thing fitting into the role where "numbers" fit into the familiar ones.
I'm of the opinion that it's reasonable to call such things numbers. However, I don't think I would ever say that out loud (other than in an opinion piece), since I would expect listeners to be confused by my usage of the word, except in cases where the word "number" is traditionally used (e.g. we say "ordinal number" versus "well-order type").
Commonly in such situations there are other words available: for example, "scalar".
A: Suppose we are considering polynomials whose coefficients are in a field $E$. If $F / E$ is a field extension and $\alpha \in F$ is transcendental over $E$, then the polynomial $p(x)$ can be considered as the number $p(\alpha)$.
