I agree with Lawrence Crowell it would be wise to stick with dimensionless
units. They're not many non integer constants in physics without a obvious
purely mathematical definition such as PI and e have but the Fine Structure
Constant is one of that rare breed of pure dimensionless numbers that
mathematicians
have never found anything special about but physicists have.

If you place 2 electrons a distance d apart they will repel each other
because they will both have a negative charge; call the energy needed to
overcome that repulsion Er and let's call the energy in one photon of light
with a wavelength of  (2PI)*d  Er.  Er/Ep is the Fine Structure Constant
(FSC), the ratio of 2 energies is obviously a pure number and is very close
to 1/137 but not exactly so, the reciprocal of the FSC obtained
experimentally is  137.035999139 plus or minus 31 in the last two digits.
It can also be calculated theoretically using Feynman Diagrams and the
result is 137.035999173  plus or minus 35 in the last two digits. Another
physical interpretation is the ratio of the velocity of a electron in the
innermost orbit of the Bohr model of the Hydrogen atom to the velocity of
light in a vacuum.

Is the Fine Structure Constant a rational number? Is it a algebraic number?
Is it a transcendental number? Nobody knows.

 John K Clark

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