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As far as I know the magnitude of constants depends on our units of measurements, so are there any units of measurements such that all the magnitude of all the fundamental constants is 1?

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    $\begingroup$ What do you mean by "all" the fundamental constants? (In other words: the electron charge $e$, the vacuum permittivity $\epsilon_0$, and the electron-electron Coulomb constant $e^2/4\pi\epsilon_0$ are all "fundamental constants", but it is obviously impossible to set all three of them to unity simultaneously. There's examples like this all around -- say, do you want to set both the Planck constant $h$ and the reduced constant $\hbar=h/2\pi$ to unity simultaneously? if not how do you choose which one to take?) $\endgroup$ – Emilio Pisanty Dec 8 '19 at 16:06
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    $\begingroup$ ... though, that said, you probably want to look into Planck units as well as the multiple versions of natural units as part of your prior research for this question. $\endgroup$ – Emilio Pisanty Dec 8 '19 at 16:12
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You can use "natural units" to set some of the fundamental constants (usually $\hbar$, $c$, Coulomb's constant) equal to one, but this fixes the values of other constants as values other than one. For example, by setting $\hbar$, $c$ and $k_e$ equal to one, you can find the elementary charge, $e$, in terms of the fine structure constant, $\alpha$.

$$\alpha=\frac{e^2k_e}{c\hbar}=e^2\approx\frac{1}{137}$$

This means you can't simultaneously set $\hbar$, $c$, $k_e$ and $e$ equal to one.

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The answer depends on the exact definition of the set of fundamental constants, as well as on the choice of base units of a given system of units.

A look at the NIST page Fundamental Physical Constants --- Complete Listing shows that they list too many fundamental constants to allow assignment of a unit value to each of them, within the International System (SI). Only a significantly reduced set of constants would allow to assign a unit value to each of them.

However, it is interesting to notice that, whatever is the set of constants, it is always possible to have all of them equal to 1, with the trivial choice of introducing a different (non-SI) unit for each fundamental quantity. For instance, one may have both electron and proton having a unit value of their mass, provided electronic mass is measured in units of mass of one electron and protonic mass in units of mass of one proton. Of course one would need a conversion factor for the two units of mass, corresponding to the ratio of protonic and electronic mass. In the same spirit the electronic charge would be $1$ in units of electronic charge and Planck's constant $1$ in units of Planck's constant. It is clear that the non unit values will be transferred from the physical constants to their conversion factors. In this sense, it would be a useless choice.

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