I referred to work by Andrew Worsley, who gives the electron Compton wavelength as 4π / n c^1½ metres, where n is a dimensionality conversion factor with a value of 1 which converts c into a dimensionless number.
Let me help you by translating that to non-baloney. The only non-baloney statement that "fixes the units", is to say that Andrew Worsley discovered the special physical constant "n= 1.00 seconds^(3/2) meters^(-5/2)" but forgot to mention it in his paper. That's a funny-looking constant, where did he get it? Nowhere, he made it up.
Let's translate Worsley's equation into a sensible unit system, then---let's measure distance in miles and time in hours.
Farsight in different units said:
I referred to work by Andrew Worsley, who gives the electron Compton wavelength as (4π / 22,442,600,000 c^1½) miles.
Farsight in different units said:
I referred to work by Andrew Worsley, who gives the electron Compton wavelength as (4π / 22.2094 c^1½) inches.
That doesn't look like a discovery any more, does it? Why bother with the 4pi and c^3/2 when you're going to throw in a made-up constant anyway?
Because what Worsley did is called
numerology. He did the following:
a) Look up the known electron Compton wavelength, in meters.
b) Look up the speed of light, in meters per second. (And, I'd guess, the fine structure constant, Newton's constant, and whatever else he thought he might need.)
c) Look up Pi, E, ln(2), and other mathematical constants
c)
Randomly throw together constants and exponents until he found a combination that happened to line up.
I just did this myself in Mathematica. Here are ten different Worsley-like equations for the low-energy limit of the electromagnetic fine-structure constant.
alpha == 2/(3*E^2*Pi^3*Log[2]^(5/2))
alpha == (2*Sqrt[Log[2]])/(E^2*Pi^3)
alpha == Log[2]/(2*E*Pi^(5/2))
alpha == 1/(E^3*Pi^2*Log[2])
alpha == 1/(E^3*Pi^2*Log[2])
alpha == 1/(E^3*Pi^2*Log[2])
alpha == (3*Log[2]^2)/(E^3*Pi^2)
alpha == Log[2]^3/(3*E*Pi^(3/2))
alpha == (2*Log[2])/(3*E^3*Pi)
Here are a pile of different Worsley-like equations for the electron mass in MeV.
me == (2*E^(3/2))/(3*Pi^(5/2)*Log[2]^3)
me == (2*E^(3/2))/Pi^(5/2)
me == Sqrt[E]/(Pi*Log[2])^(3/2)
me == Sqrt[E]/(Pi*Log[2])^(3/2)
me == Sqrt[E]/(Pi*Log[2])^(3/2)
me == 3*Sqrt[E]*(Log[2]/Pi)^(3/2)
me == (2*E^2*(Log[2]/Pi)^(3/2))/3
me == 3/(2*Sqrt[E*Pi])
me == E/(3*Sqrt[Pi])
me == (E*Log[2]^3)/Sqrt[Pi]
me == (E*Log[2]^3)/Sqrt[Pi]
me == (E*Log[2]^3)/Sqrt[Pi]
me == 3/(E^(5/2)*Log[2]^2)
me == 2/(3*E*Log[2]^2)
me == (2*Log[2])/E
me == (Sqrt[Pi]*Log[2]^(3/2))/2
me == Pi/(E^2*Sqrt[Log[2]])
me == Pi/(E^2*Sqrt[Log[2]])
me == Pi/(E^2*Sqrt[Log[2]])
me == (3*Pi*Log[2]^(5/2))/E^2
me == Pi^2/(2*E^3*Log[2]^2)
me == (3*Pi^2*Log[2])/(2*E^3)
me == (Pi^2*Log[2])/(3*E^(3/2))
me == (Pi^3*(Log[2]/E)^(5/2))/2
Oh, you don't like those dimensions? What if you don't want to use MeV? Can't I say "there's a constant n that magically fixes the units?" No need! I can ask Mathematica to find numerological coincidences in any units you like! Mathematica, what is the electron mass in pico-femtograms, please?
me == (3*E^2)/(2*Pi^(5/2)*Log[2])
me == 3/(Pi^2*Log[2]^3)
me == (2*E^(3/2))/(3*Pi^2*Log[2]^3)
me == (2*E^(3/2))/Pi^2
me == (E^(5/2)*Sqrt[Log[2]])/(2*Pi^(3/2))
me == Sqrt[E]/(Pi*Log[2]^(3/2))
me == Sqrt[E]/(Pi*Log[2]^(3/2))
me == Sqrt[E]/(Pi*Log[2]^(3/2))
me == (3*Sqrt[E]*Log[2]^(3/2))/Pi
me == 1/(2*Sqrt[E]*Log[2]^3)
me == 3/(2*Sqrt[E])
me == (2*Sqrt[E*Log[2]])/3
me == (3*Sqrt[Pi])/(E^(5/2)*Log[2]^2)
me == (Pi*Log[2]^2)/Sqrt[E]
me == (Pi*Log[2]^2)/Sqrt[E]
me == (Pi*Log[2]^2)/Sqrt[E]
me == (Pi*Log[2]^(3/2))/2
me == (2*Pi^(3/2))/(3*E^(5/2)*Log[2]^3)
me == (2*Pi^(3/2))/E^(5/2)
me == (2*Pi^(5/2)*Log[2]^(3/2))/(3*E^2)
I must be better at this than Worsley, he only found
one.