Special Exponential Functions (the Number e)

Why e is special

As interest is compounded more and more often — yearly, monthly, daily, every second — the growth factor approaches $e$. It is the limit of continuous compounding.

So $e$ is the natural choice for any quantity that grows or decays smoothly and continuously, from populations to radioactive decay.

The continuous growth formula

$A = Pe^{rt}$ gives the amount after continuous growth: $P$ is the starting amount, $r$ the rate as a decimal, and $t$ the time.

Convert the percent rate to a decimal, multiply $rt$, then evaluate $e$ to that power and multiply by $P$.

Worked examples

Try one yourself

Common questions