Fundamental units

Is it right that all units in physics can be defined in terms of only mass, length and time?

Why is it so? Is there some principle that explains it or is it just observational fact?

Which units are fundamental and which are derived is pretty much a matter of arbitrary convention, not an objective fact about the world.

You might think that the number of fundamental units would be well-defined, but even that's not true.

Take electric charge for example. In the SI system of units (i.e., the "standard" metric system), charge cannot be expressed in terms of mass, length, and time: you need another independent unit. (In the SI, that unit happens to be the Ampere; the unit of charge is defined to be an Ampere-second.) But sometimes people use different systems of units in which charge can be expressed in terms of mass, length, and time. By decreeing that the proportionality constant in Coulomb's Law be equal to 1, $$F={q_1q_2\over r^2},$$ you can define a unit of charge to be (if I've done the algebra right) $(ML^3/T^2)^{1/2}$, where $M,L,T$ are your units of mass, length, time.

Whether charge is defined in terms of mass, length, time, or whether it's an independent unit, is a matter of convenience, not a fact about the world. People can and do make different choices about it.

Similarly, some people choose to get by with fewer independent units than the three you mention. The most common choice is to decree that length and time have the same units, using the speed of light as a conversion factor. You can even go all the way down to zero independent units, by working in what are often called Planck units.

In summary, you can dial up or down the number of "independent" units in your system at will.

One more example, which seems silly at first but is actually of some historical interest. You can imagine using different, independent units of measure for horizontal and vertical distances. That'd be terribly inconvenient for doing physics, but for many applications it's actually quite convenient. (In aviation, altitudes are often measured in feet, while horizontal distances are measured in miles. In seafaring, leagues are horizontal and fathoms are vertical. Yards are pretty much always used for horizontal distance.)

It sounds absurd to think of using different units for different directions, but in the context of special relativity, using different units for space and time (different directions in spacetime) is sort of similar. If we had evolved in a world in which we were constantly zipping around near light speed, so that special relativity was intuitive to us, we'd probably think that it was obvious that distance and time "really" came in the same units.

• FYI - On the context of electric transmission wires, horizontal distances are measured in feet, and vertical (wire sag) in inches. As long different phenomena act at different directions it makes sense to differentiate the units. Commented Jun 28, 2011 at 0:45
• ""In the SI system of units (i.e., the "standard" metric system), charge cannot be expressed in terms of mass, length, and time: you need another independent unit. "" This is not true. Charge can expressed by M L and time in any system, You do not "need" a forth one. It is just a question of convenience!. On the contrary: it is possible to eliminate one of the three basic units, but then derived quantities would have more complicated dimensions. Commented Jun 29, 2011 at 16:37
• @Georg You can only do that by setting some constant to 1; and that is effectively the same thing as creating a new unit. Just that the new unit apparently disappears in equations. Commented Feb 22, 2012 at 6:28
• Came looking for this answer and can accept it but still don’t feel satisfied. The mathematics makes sense but it doesn’t make ‘physical’ sense to me. Yet. Commented Jan 21, 2018 at 1:43
• If you measure distance in seconds, you are really measuring seconds times the speed of light. You just say "seconds" as shorthand, but it's understood that $1s=1sc$ ($c$ is the speed of light). If you set $c=1$, then the numerical value doesn't change when you convert $s$ to $sc$, but the units are not meters, though they still describe distance. You use this as an example of a system with only two independent units, but as we see there are actually still three. The idea that there are only two is an illusion due to the fact that we often don't include units when we write down equations. Commented May 29, 2020 at 1:03

It is mostly by convention, since any unit could be derived from any three (independent) units. Typically we like our base units to be measurable in some sense. Distance, Time and Mass are perfectly reasonable, but not required. You can make up a unit system based on Energy, Acceleration and Density to derive all the other units.

PS. Look up dimensional analysis to figure out how to related together different units.