Exploring the Mystery of 3 Basic Units

[TamirN]

Hello everyone.
As far as I know, any unit can be expressed in terms of basic units: time, length and mass. It quite clear that we could choose them differently but in any case we get three units as the base.. And I ask: Why?
What fundamental properties of nater make it It happens that any physical unit in this crazy being can be derived from exactly 3 base unit? Why 2 isn't enough? Why not 5? What is the origin, what is behind it?

 

[Bystander]

As far as I know, any unit can be expressed in terms of basic units: time, length and mass. It quite clear that we could choose them differently but in any case we get three units as the base.. And I ask: Why?
What fundamental properties of nater make it It happens that any physical unit in this crazy being can be derived from exactly 3 base unit? Why 2 isn't enough? Why not 5? What is the origin, what is behind it?

Seven, actually; time, length, mass, temperature, amount of substance (mol), charge or current, and luminosity.

 

[TamirN]

mol is dimensionless and the rest can be expressed with time mass and length

 

[Vagn]

mol is dimensionless and the rest can be expressed with time mass and length

No they can't. How do you express charge/current and temperature in terms of time, mass and length?

 

[Dale]

any unit can be expressed in terms of basic units: time, length and mass. It quite clear that we could choose them differently but in any case we get three units as the base.

As others have pointed out, this is not generally true. In SI units there are 7 base units. In Geometrized units there is only 1. In CGS units I think that there are 3, but I am not sure.

The number of base units is simply a convention, and different systems of units use different conventions.

 

[Cutter Ketch]

As others have pointed out, this is not generally true. In SI units there are 7 base units. In Geometrized units there is only 1. In CGS units I think that there are 3, but I am not sure.

The number of base units is simply a convention, and different systems of units use different conventions.

I think I have to disagree with that. The OP was marveling at how few units it took to express all the quantities of interest in the universe (or as he/she put it "this crazy being"). I certainly can't express electricity with centimeter grams seconds. In fact it was the proliferation of nonstandard units tacked on to cgs in order to express the things that it didn't cover that led to the development of the SI units. So when cataloging the number of units it takes to express any quantity in our physical description of the universe there are many choices, many conventions, as to which particular quantities to take as base, but I don't think the number of units can be arbitrarily small. That's more like a number of degrees of freedom, and there is certainly a minimum number which span the space.

Furthermore, as there are important quantities which are expressed in the SI base units and none of them can be expressed as a linear combination of the others the minimum number must be at least seven. I would not be at all surprised to hear an example that makes it more than seven, some field of thought that still has an ad hoc unit that can't be expressed in terms of the SI units. However I can't think of one at the moment. I feel certain someone will remedy that shortly.

Editing to ad the caveat that in past cases our description of the universe has changed to make us realize things which didn't appear to be related turned out to be and the number of base units shrank. So perhaps there is som eleven dimensional theory of everything where all quantities can be expressed in a single unit

 

[mfb]

I certainly can't express electricity with centimeter grams seconds.

Yes you can, at least with Gauß' units. Charge has units $\sqrt{g cm^3} s^{-1}$, voltage is charge/cm, magnetic field strength is voltage/cm, and all other units follow in the usual way.
The German Wikipedia has a list, the corresponding English article doesn't have the same nice list in base units.

 

[Cutter Ketch]

Yes you can, at least with Gauß' units. Charge has units $\sqrt{g cm^3} s^{-1}$, voltage is charge/cm, magnetic field strength is voltage/cm, and all other units follow in the usual way.
The German Wikipedia has a list, the corresponding English article doesn't have the same nice list in base units.

I really really want what I said to be true, and I can just see that it must be true, how can it not be true, and yet it appears it isn't true. Sorry for the ignorance. I'll be over in the corner in denial for awhile trying to figure out where they hid the missing unit in the Gaussian system.

 

[Dale]

I certainly can't express electricity with centimeter grams seconds.

As @mfb answered, you certainly can. Understanding the CGS statcoulomb was one of two key turning points in my understanding of units and unit systems.

I don't think the number of units can be arbitrarily small.

I agree. It is not arbitrarily small. It is 1.

https://en.m.wikipedia.org/wiki/Geometrized_unit_system

 

[Orodruin]

How many base units you use is often a matter of convention, but there really is physics in using more than one.

Consider length and time. In relativity it becomes clear that they are only two different aspects of the same thing, displacements in space-time. As such, it really does not make sense to use different units to describe them and physicists often work in units where $c = 1$ (dimensionless). Of course you can stick to using different units to measure times and lengths, but this is nothing really strange as you can have several different units to measure the same thing (e.g., you can measure mass in kg or in pounds). In this sense, $c$ is just a conversion factor between different sets of units. When you use different units for the same basic physical dimension, this conversion factor is going to pop up in a lot of places, just like $c$ does in relativity.

Another analogy is that spatial directions are many times not really the same. For example, on the sea, a displacement of 10 fathoms in the horizontal direction would be rather inconsequential while the same displacement in the vertical direction would be rather noticeable. It makes sense to use fathoms to measure vertical distance, but horizontal distances would instead be measured in nautical miles. If you start rotating your coordinate system here you would end up with a seemingly arbitrary conversion factor between the units you use for horizontal and vertical distances, namely the ratio (1 nautical mile)/(1 fathom).