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The number and kind of units is another matter of
taste. Logically, stoichiometry should be done
by counting atoms one, two, three, but that is
often inconvenient so people use moles instead.
Similarly logically, entropy should be measured
in bits, but that is often inconvenient, so people
use Joules per Kelvin instead.
Similarly it would
be logical to measure temperature in Joules but
people commonly use Kelvins instead.
It's not a big deal; the conversion factors are
well known.
It turns out that for typical classroom measurements,
measuring distances in terms of seconds of light-travel
time is inconvenient, so people use metres and such.
Another way of saying the same thing: it depends
on whether you want to live in three dimensions
and treat relativity as a correction that crops
up when the velocity gets large ... or whether
you want to embrace four dimensions as the normal
state of affairs, and treat nonrelativistic physics
as merely the first-order approxmation, valid in the
limit of small velocity.
I find the D=4 point of view helpful for intuitive
qualitative thinking as well as for calculations.
In relativity class, you really ought to try to
embrace D=4. You can always go back to D=3 at the
end of the term if you want to. You might or might
not want to.
BTW, choosing units such that c=1 is not a problem
in practice. If a calculation tells you that the
energy is 17 kg and you want to re-express that
in some other units (e.g. Joules), it's pretty
obvious what conversion factor must be applied
(c^2 in this example).