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As far as I understand, the entire idea of dimensional analysis relies on the existence of fundamental dimensions that are independent. Because these fundamental dimensions are independent, each dimension has to appear at the same power at both sides of any equation. What is unclear to me, however, is the choice of the fundamental dimensions. Once could, for example, choose the dimensions of $length$, $time$ and $mass$ and derive the period of a pendulum simply by considering the important quantities and matching their dimensions. However, due to the existence of fundamental constants, such as the speed of light, the dimension of length and time are not independent because they can be related via $l = ct$.

Why does the dimensional analysis still work out when considering length and time as independent?

Some people include temperature, light intensity, charge, substance amount among fundamental units, however, these can be related to each other via natural constants. Which ones are the "truly" independent dimensions then and what does independence precisely mean in this context?

Upon dimensional analysis, how can we make sure that the quantities we have chosen as fundamental are indeed independent?

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It's all just a convention. Particle physicists like to put every fundamental constant to one doing calculations so that everything has dimension of some powers of energy. The fact that in the SI system some quantities are fundamental while others are derived is just because we choose them to be so. It's not really important what quantities we make fundamental, but we choose the ones so to simplify our lives in experimental measurements. What's really important is consistency.

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  • $\begingroup$ I agree that it's not the choice of a particular system of fundamental quantities what's important but consistency and different sets of fundamental quantities can work. However, how many of those quantities do you need for dimensional analysis? 1, 2, 3 or 7? Why? And how do you know they are independent? Why is length independent of time if they are related? Sorry for the question tsunami :) $\endgroup$
    – Botond
    Commented Feb 11, 2021 at 22:28
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    $\begingroup$ @Botond With the particle physics example I wanted to make it clear that the number of fundamental quantities is, itself, a convention. In particle physics we use only $eV$ as a dimension. Evidently in this system position and energy, although they might seem uncorrelated, are given by different power of the same dimension $eV$ for the former $eV^{-1}$ for the latter. $\endgroup$
    – Davide Morgante
    Commented Feb 12, 2021 at 8:04
  • $\begingroup$ Ok, so my guess is that in this system is useless to carry out dimensional analysis, am I right? Also, on the other side of the spectrum, adding more dimensions: if I considered volume as an independent dimension from length (and measured in say, gallons while length is still in meters), when doing dimensional analysis, would I still be correct? It appears that the more independent dimensions I consider in dimensional analysis, the more information I can get, but this must be wrong. I could measure areas in acres and make it independent from length, but what are the implications in dim. anal.? $\endgroup$
    – Botond
    Commented Feb 12, 2021 at 13:36
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    $\begingroup$ @Botond Quite the contrary, in fact the "God given units" are really helpful even in dimensional analysis. I reiterate: whatever you choose as "fundamental" units is not so important as consistency. BUT it's clear that certain unit systems are built to simplify things. Having independent units for volume and length would only complicate the calculations but the dimensional analysis would still be possible, just not as simple. $\endgroup$
    – Davide Morgante
    Commented Feb 12, 2021 at 15:31

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