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I was listening to a talk recently where the speaker was trying to explain how small an atom was, he was using analogies that had pretty precise numbers (a thousand billion, billion atoms in a 1mm grain of sugar; a '1' followed by 21 '0's', or as many stacked sheets of paper as it would take to get to the height of the Empire State Building) and it just got me wondering how much of quantum physics is actually empirical and how much of it is theoretical? Surely there is no way we can empirically observe something so small?

Do we arrive at these figures by following mathematics that have been formulated to explain what we so far understand of atoms and their components?

Yours inquisitively,

Gary