 Fundamental constants are some numbers with units that cannot (yet) be calculated from some physical
theory, but must be measured. 
  This
may have three possible reasons:  There is presently no theory, and there never will be a theory, that allows us to calculate fundamental constants. They have the
value they have because an act of one or more gods and/or
godesses, or they are purely random (i.e we just happen to live in an universe, where the value is what we measure.
In some other universe, or some other corner of our universe, it will be arbitrarily different).
 There is
presently no theory, but some day there will be one. Some fundamental constants will then be calculated and then are
no longer fundamental.
 There already is a theory, or at least a general theoretical framework; we just are not
yet smart enough to see the obvious or to do the numerics. Masses of elementary particles, e.g., might be
"fundamental constants" that fall into this category.

  Hotshot physicists have some ideas, which constant
might fall into which category. Speculations along this line are a lot of fun  but of no consequence so far. So
I will not dwell on this. (Of course, you may
check for yourself which one of the three possibilities you are going to embrace and thus get some idea of what kind
of person you are). 
 Fundamental physical theories usually
introduce one new fundamental constant. Mechanics (including gravitation) needs the gravity constant G, quantum
theory has Plancks constant h, statistical thermodynamics introduces Boltzmanns constant k, the special
theory of relativity (or Maxwells theory of electromagnetism which is really part of the relativity theory) needs the
speed of light c. 
  New theories
sometimes "explain" old constants of nature because they can calculate them, or replace them by something
more fundamental. Boltzmann's constant k, for example, is more fundamental than the "fundamental" gas
constant R, because it relates its number to a fundamental unit of matter (1 particle) and not to an
arbitrary one like 1 mol. 
 How many truly fundamental constants
are there? Why do they have the values they have? (Just slight deviations in the values of some constants would make
carbon based life impossible; this is where the socalled "anthropic principle" comes in). Will we eventually
be able, with a "Theory of Everything" (TOE) to calculate all natural constants? 
  Nobody knows. We run against the deepest physical
questions at this point. 
  So let's just look
at what we have. Since it is customary to list as natural constants some quantities that are actually computable from
others, we include some of these "constants" here, too (together with the conversion formula). 
Symbol and formula  Numerical value  Magnitude and unit  Remarks  Speed of light in vacuum 
c_{0}, c  2.997,924,58  10^{8}m·s^{–1}  Truly fundamental 
Gravitational constant  G  6.673  10^{–11}m^{3}·kg^{–1}·s^{–2}  Truly fundamental  Planck's constant  h  6.626,068,76  10^{–34}J·s 
Truly fundamental  4.135,6  10^{–15} eV·s  Elementary charge 
e  1.602,176,462  10^{–19}C  Truly fundamental ? Maybe not  Fine structure constant 
a = µ_{0}·c·e^{2}/2h  7.297,352,533  10^{–3}  Unitless, maybe
more fundamental than others.  Mass of a electron at rest  m_{e}  9.109,381,88  10^{–31} kg  Not truly fundamental; can be
calculated in principle  0,510 998 902  MeV 
Mass of a proton at rest  m_{p}  1.672,621,58  10^{–27} kg
 Not truly fundamental, can be calculated in principle  1.007,276,466  u  938.271,998(38)  MeV  Avogadro constant  N_{A}  6.022,141,99(47)  10^{23}
mol^{–1}  Not truly fundamental any more  Faraday
constant  F = e·N_{A}  96,485.3415(39)  C·mol^{–1}  Not
truly fundamental any more  Universal gas constant  R  8.314,472(15)  J·mol^{–1}·K^{–1}  Not truly fundamental any more  Boltzmann constant 
k = R/N_{A}  1.380,650,3  10^{–23} J·K^{–1}  Truly
fundamental  8.617,269  10^{–5}
eV·K^{–1}  Magnetic permeability of vacuum 
µ_{0} = 1/e_{0}c^{2}  12.566,370,614  10^{–7}
V·s·A^{–1}m^{–1}  Not truly fundamental 
Electric susceptibility of vacuum  e_{0} = 1/µ_{0}c^{2}  8.854,187,817  10^{–12}A·s·V^{–1}m^{–1}  Not truly fundamental 
Magnetic flux quant  P = h/2e  2.067,833,636  10^{–15} Wb  Smallest possible magnetic flux Not truly
fundamental 
