This may be a stupid question or have a pretty obvious answer, but I can't seem to find one so I'll just go ahead and post :) I was looking at some empirical data for relationships defining (abstracted) values for ionization and recomination coefficients in gases as a function of electric field strength and gas number density. I noticed that none of them had integer indices, rather they featured fractional indices correct to three decimal places. I had come across similar empirical "laws" as an undergrad studying fluid dynamics, although those seemed to be a bit more acceptable because the variable that was raised to a fractional power was always non dimensional. So for example the equation for the drag force across a flat plate (as determined from experiments) was some function of reynold's number to the power of a fractional index. As reynold's number is non dimensional, the equation would always yield newtons, not newtons to the power of some arbitrary (nonsensical) fractional index. I noticed for these equations however, the variables that are raised to fractional indices indeed have units, and the resulting equation's result is defined to have the nearest integer units. For example a/N = 3.4473 x 1034(E/N)2.985 m2 where E is the electric field (V. m-1), N is the neutral gas number density (m-3) Clearly this equation should yield fractional units but is then redefined to yield m2 How is this possible? How is it that fundamental physical relationships are always defined in terms of integer indices? Do physical phenomena happen to form perfect functional laws that feature integer indices or are these laws mere approximations that they depart from in reality? This seems absurd so I'm guessing it has something to do with the way that our arbitrary mathematical constructs are formed, or perhaps how we define the dependent variables involved. (I'm thinking of 1/2kT2, where T is defined to have a functional relationship to energy involving an integer index, but perhaps there's a better example). As an unrelated sidenote, what makes e and pi the values that they are? Physical constants are arbitrary, but these constants are ratios of abstract concepts. Changing our number system would change their values superficially, but they would still represent the same quantity.