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Arguably the most well-known and used system of units is the SI-system. It assigns seven units to seven ‘fundamental’ quantities (or dimensions). However, there are other possible options, such as Gaussian units or Planck units. Until recently, I thought that these different systems differed only in scale, e.g. inches and metres are different units, but they both measure length. Recently though, I discovered that it is not simply a matter of scale. In the Gaussian system for example, charge has dimensions of $[mass]^{1/2} [length]^{3/2} [time]^{−1}$, whereas in the SI-system it has dimensions of $[current] [time]$. Also, I have always found it a bit strange that mass and energy have different units even though they are equivalent, but I find it hard to grasp that a quantity can be ‘fundamental’ in one system, and not in an other system.

Does this mean that all ‘fundamental’ quantities are in fact arbitrary? Would it be possible to declare a derived SI-unit fundamental, and build a consistent system with more base units? What is the physical meaning of this?

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Have you considered Natural Units? – Alfred Centauri Jun 24 '13 at 20:45
Yes, Planck units are an example of natural units. The point is, I do not understand how dimensions of quantities can be different in different systems of units. – Ruud Jun 24 '13 at 20:49
the two systems you mentioned are different because Gaussian units incorporate many of the natural constants into the units. Charge is in terms of mass, length, and time only because the constant $\epsilon_0$ has been applied to it (essentially) – Jim Jun 24 '13 at 20:54
This papaer could be very interesting for you – Noldig Jun 24 '13 at 20:55
That paper is great. Here's an illustrative quote from it: "The SI might be useful from the point of view of technology and metrology, but from the point of view of pure physics four out of its seven basic units are evidently derivative ones. Electric current is number of moving electrons per second. Temperature is up to a conversion factor" – Ehryk Jun 24 '13 at 21:32

The key difference is the $ \frac{1}{4\pi\epsilon_0} $, with $ \epsilon_0 $ in the SI formulation of charge, being vaccuum permittivity with units $ (charge)^2(time)^2 (mass)^{−1}(length)^{−3} $. This satisfies the unit cancellation, and in the SI system makes the electric constant $ \mu_0 $ and $ \epsilon_0 $ now derived units. (See Vacuum Permittivity or SI Unit Redefinition)

For example, Coulomb's law in Gaussian units appears simple:

Gaussian Formula

where F is the repulsive force between two electrical charges, Q1 and Q2 are the two charges in question, and r is the distance separating them. If Q1 and Q2 are expressed in statC and r in cm, then F will come out expressed in dyne. By contrast, the same law in SI units is:

SI Formula

where $ \epsilon_0 $ is the vacuum permitivity, a quantity with dimension, namely (charge)2 (time)2 (mass)−1 (length)−3. Without $ \epsilon_0 $ , the two sides could not have consistent dimensions in SI, and in fact the quantity $ \epsilon_0 $ does not even exist in Gaussian units. This is an example of how some dimensional physical constants can be eliminated from the expressions of physical law simply by the judicious choice of units. In SI, $ \frac{1}{\epsilon_0} $, converts or scales flux density, D, to electric field, E (the latter has dimension of force per charge), while in rationalized Gaussian units, flux density is the very same as electric field in free space, not just a scaled copy. Since the unit of charge is built out of mechanical units (mass, length, time), the relation between mechanical units and electromagnetic phenomena is clearer in Gaussian units than in SI. In particular, in Gaussian units, the speed of light $c$ shows up directly in electromagnetic formulas like Maxwell's equations (see below), whereas in SI it only shows up implicitly via the relation .

- Wikipedia: Gaussian Units

Yes, I would argue that 'fundamental quantities' are indeed arbitrary, as are many of our choices, such as base-10 number systems. This is illustrated well on the Golden Record we put on voyager spacecraft, for decoding by other intelligent life; we show how fast to spin the record by relating time units in the fundamental transition of the hydrogen atom:

Golden Record Instructions

I'd then add that we have tried to make them as least-arbitrary (to us) as possible, but there's no reason that some other intellegence would have different 'fundamental unit' definitions and scalings, or whatever 'arbitrary' units they came up with. We could use $ (time)^{-1} $ or 'period' as our fundamental timing unit, and change all the other derived units to follow, if we wanted.

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I understand that one can replace any base unit by an other unit, and re-define everything in terms of the new unit (it always seemed more logical to me to take the Coulomb as a base unit, and define Ampère in terms of it), and in this sense base units are arbitrary, but this does not change the dimensions of the quantities they measure. As a reformulation of my question, is the number of base units arbitrary? (E.g. it seems that one can remove the base unit for current and define it in terms of the other units.) – Ruud Jun 24 '13 at 21:36
Do you mean for the same system? One could postulate a civilization that is pre-relativity, and doesn't have $c$ defined. If you mean to also end up with the same result, I.E. express all of the SI system's units (and only those) then I think the answer is still yes. Could I not redefine length to use two 'arbitrary' new base units, like 'velocity of planet around main star(s)' and 'planetary rotation time'? What's your definition of 'base unit'? – Ehryk Jun 24 '13 at 21:42
To cut to the root of your question, I'm fairly certain that a given civilization or unit systems choice of what are 'fundamental units' are completely arbitrary. "A set of fundamental units is a set of units for physical quantities from which every other unit can be generated." (wikipedia) – Ehryk Jun 24 '13 at 21:49
Note that it doesn't specify "the smallest set", perhaps you mean 'base units' to be the "smallest complete set of fundamental units"? – Ehryk Jun 24 '13 at 21:51
I meant the SI base units, the set of units for physical quantities from which every other unit can be generated. I understand that the scales of these units, and which ones you assume to be ‘fundamental’ is arbitrary, but what about the dimensions of the quantities they measure, e.g. length, time, mass, etc.? – Ruud Jun 25 '13 at 6:38

The Si - Units are just a definition which are related by euqations. Take for example the speed of light $c \approx 3\cdot 10^8\frac{m}{s}$. Now what does $\frac{m}{s}$ mean? You can take it as a parameter that is connected to other units by equations like the famous $E = m c^2$. Since only the equation is important and you have to define your unit somehow you could also say that $c = 1$.

What I did here is nothing else than to set $$\frac m s = \frac 1 {3\cdot 10^8}$$ You can always do this for the first unit you change, but you have to be careful if you change a second unit as those units may be connected by an equation.

Take again $E = mc^2 = m$ where I have set $c = 1$. Now there is one more independent unit: Either mass or energy which you also can choose as you like. As you can see there is an infinite number of possibilities to choose the units, but as people have to communicate it is a very good advice to keep on the standard units in the different fields of science.

An important note is the following: We have set $c = 1$ this means, that length has the same unit as time. Take as an example a star $\Delta x = 100 c \cdot s$ away. Here $c \cdot s$ are light seconds. As we have set $c = 1$ you can clearly see that $\Delta x = 100 s$ which is not very intuitive but you have to keep in mind that $c = 1$ and with this you can always change from meters to seconds with $s = 3\cdot 10^8 m$ in this system.

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