It’s been little over 100 years since we discovered radioactivity. How then do we know the half life of elements that have half lives in hundreds plus years?
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The half-life isn’t the time it takes one atom to decay. Decay is randomly happening all the time, even in relatively stable isotopes.
The half-life represents the average rate of decay. It is the time, on average, that it takes for half of the atoms in a pure sample to decay.
If you have, say, a 2kg sample of U-238, there are a *very* large number of atoms in that sample (many orders of magnitude more than a billion) and at least some of them will be decaying – we can detect the radiation from those decay events that do occur effeciently enough to calculate the time it would take for half of that sample to decay.
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