There are a few dimensionless quantities in physics. For instance, if e is the charge of the electron, \hbar is the Planck constant, and c the speed of light, then as we have already noted there is a dimensionless combination \frac{e^2}{\hbar c} \approx \frac{1}{137}. Unfortunately, this is one of the few natural dimensionless constants in physics that is of a reasonable order of magnitude. A primary motivation in modern theoretical physics is to understand the “unnaturalness” of other constants. For example, the Planck mass is 10^{19} GeV. In other words, the mass of the proton in the natural units of the universe (i.e., the Planck units) is m_p/M_{planck}=10^{-19} . This is a tiny dimensionless number which is fundamental to physics and certainly not order 1!
Interestingly, the natural mass scales fall into roughly three groups, separated by about 30 orders of magnitude altogether, i.e., factors of 10^{30} . In Planck units in logarithmic scale we have
