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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:


Can we consider polynomials as numbers? At least in some specific sense?

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The set of polynomials along with arithmetic $\times, +, -$ form a ring: See Wikipedia page on polynomial rings. – user2468 Aug 20 '12 at 1:32
I don't see any structure in the infinite series for $e$ that is similar to the structure of a polynomial. What in the series corresponds to $t$ in the polynomial? – Gerry Myerson Aug 20 '12 at 1:58
That they are both a sum of products of things is a rather superficial connection between the two in my opinion. Polynomials are finite, formal, algebraic gadgets, whereas numbers are more concrete (and in this case the real number has an infinite expansion which involves an infinite limiting process). – anon Aug 20 '12 at 3:15
If one is in the mood, one can consider "numbers" to be very special polynomials. – André Nicolas Aug 20 '12 at 4:44
Search youtube for polynumbers, Math foundation course by Norman Wildberger. – Arjang Aug 20 '12 at 7:40
up vote 3 down vote accepted

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".

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I think your view of polynomials as extensions of the number concept is not an uncommon one. If I find a cite I will post it here. – MJD Aug 20 '12 at 4:02

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.

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I wonder if it's worth mentioning that the mapping $\phi_e(p)$ which takes a polynomial $p$ to the number $p(e)$ is a ring homomorphism. – MJD Aug 20 '12 at 1:44
Can you comment or point me something with this distinction? – Voyska Aug 20 '12 at 1:44
Tangent: MJD has a good question relating to how the evaluation map from formal polynomials to the base ring is a ring homomorphism if and only if the ring is commutative. – anon Aug 20 '12 at 3:33

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 $p(\alpha)$.

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