Expert: Scott A Wilson - 10/27/2011

Question
100(.5)^n=0

I'm finding the rate of decay. N is the number of years. The equals 0 is when there is no more isotope left. What is n?

Answer
This is not possible, for 0.5^n is always positive, and 100 times a positive is still a positive.

If we only wanted 2 places accuracy, it would be 100*(0.5)^n < 0.005.
Dividing by 100 means 0.5^n < 0.00005.
Since 0.5^n = 1/2^n = 2^(-n), we could then take log base 2 and get -n = log(0.00005), base 2.
This means n has to be at least 15.  Now that 100*0.5^15 = 0.003051758,
and that is less than 0.005.

If we wanted 4 places accuracy, it would be 100*(0.5)^n < 0.00005.
This would make 1/2^n < 0.0000005.  In this case, n would be 21, for 100*0.5^21 = 4.76837E-05,
and that is small enough.

Depending on your computer, if n = 1022, then =100*0.5^n gives 2.2251E-306,
and if n is increased by 1, we have n = 1023, the value in Excel of =100*0.5^1023 is 0 on my PC.
This does not mean it is truly 0, but that is all the more accurate Excel is.
Note that 2^1023 = 8.9885E+307, so 100*0.5^n = 100/2^n = 100/8.9885E+307,
and that is close, but is still not 0.