Is the fine structure constant a rational number? Since the fine structure constant (denoted alpha) is a pure real number, it just occured to me to ask if it is a rational number or not.
 A: The fine structure constant is given as: $$\alpha = \frac {k_{e} e^2} {\hbar c^2}$$
Immediately we have a problem in determining the rationality or otherwise of $\alpha$. The Coloumb constant, Planck constant (maybe not?) and speed of light are all either exact numbers or pre-defined. Since the elementary charge $e$ is an empirically derived constant we can only measure a finite number of digits of this number. This is true for two reasons: First any measurement device will have $\it{some}$ degree of imprecision. Second, the uncertainty principle tells us we cannot measure electron motion+position simultaneously with perfect precision anyway.
Taking this into account we can only know a finite number of significant figures of $\alpha$ and so it must be rational empirically.
As @CuriousOne points out though, this is a rather meaningless concept.
A: As other people have stated, we are currently getting this quantity experimentally, so thus far this is not an appropriate question.
The origin of the value ~1/137 remains unexplained; There is currently no theoretical explanation for why is has the value that is has. In the future, physicists might be able to explain its origin, perhaps by deriving it from other more fundamental constants. Then we would be able to answer your question.
Contrast this with, say, the gyroscopic factor, which characterizes the magnetic moment of quantum particles. There is a theoretical derivation for this dimensionless quantity which can be derived from the theory of quantum electrodynamics.
Someday, we hope, the same will be true of $\alpha$.
