Unexpected approximations which have led to important mathematical discoveries On a regular basis, one sees at MSE approximate numerology questions like 


*

*Prove $\log_{{1}/{4}} \frac{8}{7}> \log_{{1}/{5}} \frac{5}{4}$, 

*Prove $\left(\dfrac{2}{5}\right)^{{2}/{5}}<\ln{2}$, 

*Comparing $2013!$ and $1007^{2013}$
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.
 A: 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. 
A: 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.
A: 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.
