What has Euler's number $e$ to do with exponential decay? I know how to derive the formula for "quantity at time $t$" for some decaying materials. You can see the derivation here. But, what I don't get is that what the number $e$ is doing here? We get the value of $e$ from the equation of compound interest. Compound interest and decay (like, radioactive decay) are two different things. Is there any intuitive way to see what the Euler's number doing here?
 A: Compound interest and radioactive decay both vary exponentially with time. That simply means that in any set period of time, the value changes by the same fractional amount.
Any exponential function can be written using any base.  If the number of remaining atoms in a sample is given by $A * e^{-Bt}$, then it is also given by $A * 2^{-Ct}$, for the appropriate value of $C$.
Regardless of the base you choose, the number $e$ pops out when you calculate the rate of change of the exponential function (which is another exponential function). So, assuming you are interested in, say, both the number of atoms and how fast that number is changing, you will be stuck with an $e$ somewhere or other anyway. Choosing it as the base eliminates a constant that would otherwise appear in the expressions.
A: By definition, the natural log base $\mathrm{e}$ has the special property that
$$ \tfrac{\rm d}{{\rm d}t} \mathrm{e}^t = \mathrm{e}^t $$
This leads us to understand that all solutions to
$$ \tfrac{\rm d}{{\rm d}t} f(t) = f(t) $$ have basis functions $f(t) \propto \mathrm{e}^t$.
A subset of these problems are that of
$$ \tfrac{\rm d}{{\rm d}t} x(t) = -a\, x(t) $$ which have solutions of the form $x(t) = X \mathrm{e}^{-a t}$
This is the basic exponential decay formula, and the interpretation of the differential equation is that the amount decaying is proportional to the amount that exists at any time.
A: Compound interest is exponential decay when you examine it closely.
Look at e in the formula for compound interest: (1 + 1/n)^n
Look at the fraction 1/n. The larger n gets, the smaller the fraction gets. In fact, it gets exponentially smaller. Exponential decay.
In the infinite sum for the compound interest equation above, the larger n gets, the closer it gets to e. If you let n run to infinity, the answer would converge to e. An easy way to see this is to set principal to $1.00, interest rate to 100%, and compounding periods to fractions of one year i.e. for compounded every six months would be (1+ 1/2)^2. Then try higher and higher numbers for n. The return on principal plus principal approaches e. Exponential decay of return.
In fact, when you graph the compound interest formula, it looks like a reflection of e^x ( See Wolfram Alpha on the constant e).
Euler’s number has to do with exponentials. All constants to a time exponential, have a derivative that has a proportionality constant.  When the proportionality constant is the number 1, the constant is e.
However, it should be understood that the compound interest formula has a limit of e, whereas, the function e^-x has a limit of zero. And it should be understood that many infinite series can converge to a real number. To understand why e (Euler’s number) is special, see YouTube by 3Blue1Brown called “What’s so special about Euler’s number e?”
“If you invest your money, the rate at which it grows is proportional to the amount of money there at any time. In all of these cases where some variable’s rate of change is proportional to itself, the function describing that variable over time is going to look like some kind of exponential. And even though there are lots of ways to write any exponential function, it is very natural to choose to express these functions as e to the power of some constant times t since that constant carries a very natural meaning. It’s the same as the proportionality constant between the size of the changing variable and the rate of change.”—Grant Sanderson
But if you are asking “Why?” e shows up in biology, math(s), compound interest, radioactive decay, etc., that is like asking why pi shows up in population growth and many other functions when it comes from a circle formula.  Most of Science can’t answer Why? They can measure observations, predict from observations with mathematical principles, but cannot yet answer “why?” IMHO, someday they will.
The more interesting question is if radioactive decay has a predictable half-life, how can it be spontaneous i.e. meaning “random decay” without a pattern in the individual atom. Quantum Physics cannot yet answer this. They can measure decay in half-lives, but they also can mathematically explain spontaneous emission. How does something random form a pattern?
BTW, physics cannot really be separated from mathematics, so your question is a physics question.
