I understand the idea of swapping from unit systems, say from $\mathrm{m\ s^{-1}}$ to $\mathrm{km\ s^{-1}}$, but why can we just delete the units altogether?
My question is: what exactly are we doing when we say that $c=1$?
Answer
All we're doing is using a set of units where certain quantities happen to take convenient numerical values. For example, in the SI system we might measure lengths in meters and time intervals in seconds. In those units we have $c = 3 \times 10^8\ \text{m}/\text{s}$. But you could just as well measure all your distances in terms of some new unit, let's call it a "finglonger", that is equal to $2.5 \times 10^6\ \text{m}$, and time intervals in a new unit, we'll call it the "zoidberg", that is equal to $8.33 \times 10^{-3}\ \text{s}$. Then the speed of light in terms of your new units is $$ c = 3 \times 10^{8}\ \text{m}/\text{s} = 1\ \frac{\text{finglonger}}{\text{zoidberg}} .$$ The units are still there – they haven't been "deleted" – but we usually just make a mental note of the fact and don't bother writing them.
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