Radioactive decay works like continuous compounding in reverse. A radioactive material will dissipate with time, its molecules breaking down into molecules of other substances. If you have a certain amount present at a given time, it is found that the proportion that is lost depends only on the type of material and the time elapsed.

If you start with 100 g, then at the end of 1 day there will remain 100k g, where k is a constant, between 0 and 1, depending only on the material. After n days,  the amount remaining is 100kn g. This is exactly the same formula as continuous compounding, with e^r = k.

Of course, if er is to be smaller than 1, r must be negative. So radioactive decay is an example of exponential growth with a negative exponent.

The half-life of an element is the time it takes for the amount of it present to halve. For example, if you have 500 g with half-life 1 year, there will be 250 g after 1 year, 125 after 2 years, and so on. After n half-lives amount A decays to A/2n.

Sample Problem 1.1 An artificial element has a half-life of 1 h. You have 450 g. Approximately how long will it take until only 50 g is left?

Solution. You want 450/2n = 50.

n = 3 : 450/23 = 450/8 = 56.25,

n = 4 : 450/24 = 450/16 = 28.125,

so the approximate answer is: a little over 3 h.