# The meaning of the natural number, e

## Growth is easier to think about than decay

Let’s change our ‘rate of decay’ equation so it’s simpler to use.

Instead of dN/dt = -λN we'll use dN/dt = N;

This is no longer the equation for radioactive decay. It's the equation for exponential growth. It says ‘the more you have, the faster you get it’.

*dN/dt = N* says that at every value of t, the gradient, dN/dt, is equal to N.

What function has the property that the gradient equals the value?

Each time-step the *current* total grows by the same proportion. So a good guess is to raise some number to the power of time.

N = some number^{t}

The time tells you the number of steps and you'll get a better estimate if you use lots of very little steps rather than a few big ones.

It turns out that N = e^{t} has this property that the gradient dN/dt at any time is always the same as the value, e^{t}.

## From growth to decay

We've looked at dN/dt = N but we want dN/dt = -λN.

So instead of N = e^{t} we want N = N_{0}e^{-λt} .

The first thing to notice is that the negative sign makes it exponential decay (the less you have the slower you lose it) rather than growth.

In most cases the gradient dN/dt is *proportional* to the value, N, rather than equal to it and that's where our decay constant, λ, comes in.

λ has the effect of changing the shape of the graph so that the half-life changes. N_{0 }changes the size of the graph so that there are the appropriate number of undecayed nuclei at each time.