How can integration be used in deriving radioactive decay formula?

Question

I recently learnt the derivation of radioactive decay formula and I am quite surprised about using integration to derive the formula.

But $N$ (the number of atoms) can only be discrete numbers (like 1,2 & not 1.5 or.9).
But won't using integration include all the continuous values of $N$?
So instead of using integration, shouldn’t some kind of summation be used to include the discrete values only?

This answer says something like this about the use of summation instead of integration. https://math.stackexchange.com/q/1509235/

$$ \begin{align} \frac{dN}{dt} & = - \lambda N \\ \end{align} $$ hence, $$ \begin{align} \int \frac{dN}{dt} & = - \lambda \int dt \\ \\ \ln{N} & = - \lambda t + C \end{align} $$ If the initial number of nuclei is $N_0$ and $N = N_0$ when $t = 0$
then (i becomes, $$ \begin{align} \ln N_0 = C \end{align} $$ substituting for C into (i $$ \begin{align} \ln{N} & = - \lambda t + \ln{N_0} \\ \ln{N} - \ln{N_0} & = - \lambda t \\ \ln({ \frac{N}{N_0} }) & = - \lambda t \\ \Rightarrow \frac{N}{N_0} & = e^{-\lambda t} \\ \Rightarrow N & = N_0 e^{- \lambda t} \end{align} $$

Answer 1

Since the atoms decay at random, $N$ should be properly understood as the expected number of atoms left: $$N = \sum_{n=0}^\infty n\, P(\text{there's $n$ atoms left})$$ Since probabilities are not integers, $N$ doesn't need to be an integer either, and it satisfies the continuous differential equation.

Alternatively you can use the differential equation to calculate the probability that any particular atoms has not yet decayed after time $t$ to get $$ P(\text{atom has not yet decayed}) = e^{-\lambda t}$$

The actual number of atoms left at any given point may differ from $N$ but for large amounts of atoms the difference will be relatively small.

Answer 2

If you restrict $N$ to be discreet, then you can also not write $dN/dt = -\lambda N$, as the derivative implies a continuous function of time. You should rather think of $N$ as the average number of atoms in an ensemble. We prepare a large number $M$ of identical setups, each containing $N_0$ atoms at $t=0$, and measure the average number of atoms in these ensembles at each point. If $M$ is large enough, we can treat $N$ approximately as a continuous function. (at the end, if you want, we can think of all the ensembles as one large setup, and it still works)