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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A half-life is a mathematical result if a percentage of the atoms decays in a unit of time, so the number you looses depends on the number you have.
It gets simpler to understand if you look at growth and the time it takes to double If you have a bank account where you put in 10000 today and after a day you get 10005 because of interest. You can use that information to determine how long it takes to double of the interest in constant.
10005/10000= 1.0005 which is a 0.05% interest per day. So you have 10000*1.0005 after one day 10000*1.0005*1.0005 after two. You can rewrite it as 10000*1.0005^n for n day.
Now the problem is to solve 10000*1.0005^n = 10000*2 => n≈1386.64
You simplify it and remove the 10000 part and just solve 1.0005^n = 2 This simplification shows the time to double the money is independent of the amount of money you start with.
1386.64/365 = 3.799 years which is a bit less than 3 years and 10 months.
0.05% interest per day is equal to 20% interest per year.
So by just observing the interest during the day, we can determine the time it takes for the money to double.
You can do the say with decay. You start with 10000 atoms and after one day 10 have decayed (10000-10)/10000 =0.999. Then the problem is to solve 0.999^n =1/2 =>n≈692.8 which is a bit less the 1 year and 11 months.
This calculation for decay is a bit of a simplification, it gets better if you calculate the instant rate and not look at it in the resolution of just days. The point is still shown you can calculate the half-life from the current activity. 10000 atoms is also a very small amount and real observation of most elements will have billions of atoms.
So you just need to know the number of radioactive atoms you have and the number of decays you get right not to calculate how long the half-life is
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