Can we get the values of $G$, $h$ and $c$ to be numerically equal if we use a convenient system of measurement?

The speed of light in vacuum is approximated to be $$3×10^8\ ms^{-1}.$$ But, if we change the units, we can get a different number. For example, it won't be $$3×10^8$$ if we used $$ft\,s^{-1}$$ instead of $$ms^{-1}$$.

In the SI system, however, we know that the numerical value of the speed of light, Planck's constant, and the Universal Gravitational constant are different.

Is it possible that we get those constants to be numerically equal if we introduced a certain system of measurement which allows it to happen? Can such a system of unit even exist in the first place?

• Yes, these units are called natural units, though usually one choose $\hbar = G = c = 1$, not $h$. – knzhou Jul 8 at 0:03
• – AccidentalFourierTransform Jul 8 at 0:03
• It's not a measurement, it's a definition. – The Photon Jul 8 at 0:30
• The speed of light is exactly 299,793,458 meters per second, because we have defined the meter to make it have this value. You can define units to make it have any value you want. This is why theoretical physicists just set it to 1. – G. Smith Jul 8 at 0:31
• The speed of light used to be a measurement, now it is exact and used to define a meter, if that caused any confusion. – user47014 Jul 8 at 1:32