Suppose the radioisotope has half-life of \tau and initial number of N_0 , the number of isotopes at give time is

\displaystyle N(t) = N_0 2^{-\frac{t}{\tau}}

The number of decay at a given time, measured with a time period \Delta t is

\displaystyle D(t, \Delta t) = N(t - \Delta t) - N(t)

The rate of number of decay at a given time is

\displaystyle R(t) = \lim_{\Delta t \rightarrow 0} \frac{N(t- \Delta t) - N(t)}{\Delta t} = - N'(t) = \frac{\log2}{\tau} N_0 2^{-\frac{t}{\tau}} = \frac{\log2}{\tau} N(t)

or

\displaystyle N(t) = \frac{\tau}{\log2} R(t)

This, the rate of number of decay is always proportional to the number of radioactive isotopes, therefore, by measuring the radioactivity, the number of radioactive isotopes can be deduced.

For example, suppose we have a sample of tritium with half live of 3.89 \times 10^8 sec. When we measure the sample, we got 100 beta decay in 1 min, that means, the radioactivity is 1.67 decay per second. Therefore the number of isotopes is 9.14 \times 10^8 tritium. Notice that, this number is the number of isotopes at the end of measurement.

For very short half-life, say, few mili-sec, if we collect the number of decay in a few seconds, the number of isotopes changed a lot and the simple ratio \frac{\tau}{\log2} is no longer accurate.

\displaystyle \frac{D(t, \Delta t)}{N(t)} = \frac{N(t - \Delta t) - N(t)}{N(t)} = 2^{\frac{\Delta t}{\tau}} - 1

and

\displaystyle \lim_{\tau \rightarrow \infty} \left( 2^{\frac{\Delta t}{\tau}} - 1 \right) \frac{\tau}{\log 2} = 1