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On a regular basis, one sees at MSE approximate numerology questions like

or yet the classical $\pi^e$ vs $e^{\pi}$. In general I don't like this kind of problems since a determined person with calculator can always find two numbers accidentally close to each other - and then ask others to compare them without calculator. An illustration I've quickly found myself (presumably it is as difficult as stupid): show that $\sin 2013$ is between $\displaystyle \frac{e}{4}$ and $\ln 2$.

However, sometimes there are deep reasons for "almost coincidence". One famous example is the explanation of the fact that $e^{\pi\sqrt{163}}$ is an almost integer number (with more than $10$-digit accuracy) using the theory of elliptic curves with complex multiplication.

The question I want to ask is: which unexpected good approximations have led to important mathematical developments in the past?

To give an idea of what I have in mind, let me mention Monstrous Moonshine where the observation that $196\,884\approx 196\,883$ has revealed deep connections between modular functions, sporadic finite simple groups and vertex operator algebras.

Many thanks in advance for sharing your insights.

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    $\begingroup$ I've rediscovered many known identities of well known constants/functions with series involving zetas and logarithms and combinatorical constants by use of approximations (over the truncated series), often even occuring as divergent sums using Euler-summation... unfortunately these are no more "important mathematical developments" in our century (so this comment surely isn't qualified to become an answer, sorry, but I think is worth to be mentioned) $\endgroup$ – Gottfried Helms May 13 '13 at 18:29
  • $\begingroup$ Exponential convergence of the optimal rational approximations to the square root function on an interval led us to a discovery of a special continued fraction: onlinelibrary.wiley.com/doi/10.1002/… $\endgroup$ – DVD May 19 '13 at 4:00
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    $\begingroup$ This is not exactly what you are looking for, but some of the answers in the thread come close. But I'm not sure about the "important mathematical developments"-part though. $\endgroup$ – 77474 May 20 '13 at 18:35
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    $\begingroup$ Not exactly what you're asking, but Noam Elkies has a short explanation of why $\pi^2 \approx 10$: math.harvard.edu/~elkies/Misc/pi10.pdf $\endgroup$ – Michael Lugo Jun 11 '13 at 18:13
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    $\begingroup$ We can start with oeis.org/A002072, and look at the two highest consecutive numbers that are both 19-smooth. From there, we get the best known solution of one form of the ABC conjecture, and it turns out to be equivalent to $\sqrt{\sqrt{9.1}} = 33/19$ $11859210 ~ 11859211 => 7×13×19^4 ~ 2×3^4×5×11^4 => 91×19^4 ~ 10×33^4 => 9.1 ~ 33^4/19^4 $ $\endgroup$ – Ed Pegg Jun 20 '13 at 17:14
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The most famous, most misguided, and most useful case of approximation fanaticism comes from Kepler's attempt to match the orbits of the planets to a nested arrangement of platonic solids. Fortunately, he decided to go with his data instead of his desires and abandoned the approximations in favor of Kepler's Laws.

Kepler's Mysterium Cosmographicum has unexpected close approximations, and they led to a major result in science.

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    $\begingroup$ I like this answer! In particular, because it approaches the question from an unexpected angle. It is also interesting to note that some people continue in the spirit of initial Kepler's attempts, $\endgroup$ – Start wearing purple Jun 20 '13 at 19:55
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    $\begingroup$ @O.L. and Ed : There is an excellent story about Kepler's own point of view in Koestler's book : 'The Sleepwalkers'. Kepler didn't seem really impressed by his three laws (rather hidden in his papers and rediscovered by followers like Newton...) even if he used them for the edition of his new astronomical tables (a rather exhausting and long work). In his late years his mathematical explorations and interest seem rather related to his platonic solids... $\endgroup$ – Raymond Manzoni Jun 22 '13 at 16:54
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    $\begingroup$ Raymond -- that makes it even funnier. $\endgroup$ – Ed Pegg Jun 24 '13 at 14:05
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Not sure if this is unexpected or so but I think the fast inverse square root is kinda cool. Don't think it lead to any mathematical developments though it's been implemented more widely since.

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Following on from @leonbloy's comment to the OP: Pythagoras's system of tuning is based on octaves, of frequency ratio 2:1, and perfect fifths, of frequency ratio 3:2. (Pythagoras observed that an interval whose frequency ratio is the ratio of small integers sounds pleasant.) Thus the frequency ratio of any interval in this system is the ratio of two 3-smooth integers. For example, the tone, 9:8, and the ditone, 81:64. The tone sounds pleasant, but the ditone does not, and one reason is that the integers are somewhat large.

Some later music theorists, such as Archytas and Zarlino, used the major third 5:4 instead of Pythagoras's ditone of 81:64. 5:4 sounds pleasanter because the numbers are smaller. Substituting 5:4 for 81:64 is practicable only because they are so close in size: the difference between them is the syntonic comma 81:80. 81 & 80 are the largest 2 consecutive 5-smooth integers.

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