The process of radioactive decay has a particular mathematical property. The rate of radioactive decay
is proportional to the number
of undecayed atoms present. We can write
where
is the decay rate,
is the number of undecayed atoms present
is the decay constant.
We can solve this differential equation using a method called separation of variables:
To fully solve the original differential equation we have to find
When
the initial number of undecayed atoms. Substituting these into the equation
gives 
Hence
The half life
is the time taken for half the atoms to decay ie for
to fall to
We can find
to
is the same as the time taken to fall fromto
etc. The graph below illustrates this.