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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.
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    $0$ and $1$ are rather interesting constants, and though they may be considered "small integers" their interest is not merely combinatorial. – 2017-01-02
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    Even the numbers you list $\pi, e, \gamma$ [MathWorld](http://mathworld.wolfram.com/topics/Constants.html) lists under "Number Theory", so I think you should be more specific about what areas of math you want to include or exclude (in particular, different people will have different ideas what analysis is). – 2017-01-02
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    $\gamma$ is not proved as irrational – 2017-01-03
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    I suspect that most constants that were originally considered in analysis and turned out to be rational actually permit a definition in purely algrabraic/combinatorical terms, and hence aren't classified as “analysis constants”. – 2017-01-03
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    What fractions aren't "common"? What exactly differentiates $3/2$ from $1728/401$? I would argue that all such fractions are basically uninteresting, in the sense that they fit into a framework that we know. Irrationals can have questions asked about them, like "how can we compute this efficiently?", "is is algebraic?", "can we express this in terms of some special functions like $\Gamma?$" That said, do the Bernoulli numbers count as "constants of analysis"? They are defined by coefficients in certain Taylor series, are rational, and are certainly of interest to many people. – 2017-01-03
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    Do fractal dimensions count? Rational constants are infrequently named simply because they are easily described as a ratio of integers, whereas irrational constants (except for simple algebraic ones and similar) have no concise description, which is why it's useful to invent symbols for them when they do arise. – 2017-01-03

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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.

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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.

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    "We all know that Gödel numbers are interesting." Speak for yourself, personally I find Gödel numbers very boring. Like looking at the hex dump of a JPEG of the Mona Lisa... – 2017-01-03