Are any constants (from analysis) rational?

Question

There seems to me to be a meta-principle of analysis that all the "interesting" mathematical constants - $\pi$, $e$, $\gamma$, Bessel function zeros, etc. - are irrational (even transcendental). I'm wondering if there are any counterexamples to this principle - are there any "interesting" constants arising in analysis that are rational?

What constitutes an interesting mathematical constant is of course murky. I will propose the following guidelines to make the question somewhat more well-defined:

1. Small integers are excluded. (These come up all the time for algebraic and combinatorial reasons.) Common fractions like 3/2 should probably be excluded on the same grounds.
2. To be eligible, a constant should be "natural" in the sense that its definition does not depend on arbitrary choices for some other parameters.
3. I'm restricting this question to constants of analysis because some other branches of math, like group theory and combinatorics, have an abundance of large integer constants (e.g. order of the Monster Group). To qualify as a constant of analysis, it should not be derivable from purely algebraic or combinatorial principles.

Answer 1

You might look here to see which of the listed examples relate to calculus.

Edit: In the link above, $\frac{32}{27}$ related to the closure of the set of all real zeros of all chromatic polynomials of graphs seems a good candidate to me.

Answer 2

There exists interesting natural (hence rational) constants in analysis that fulfill the conditions stated above.

Proof

Let $S$ be the natural $\lceil s\rceil$. Since $s$ is small, $S$ is also small. We can all agree that the number of interesting theorems in analysis is not small, therefore exists a $K>S$ with (analysis theorem $K$ is interesting) and $K$ not small. We all know that Gödel numbers are interesting. If mathematical theorems are ordered by their Gödel numbers and enumerated, we know that (Gödel number of theorem $n$) $\geq$ $n$. It follows that (Gödel number of analysis theorem $k$) $\geq$ $k$. Hence we know that (Gödel number of analysis theorem $K$) is interesting and not small, furthermore strongly analysis-related and rational. Q.E.D.