Never forget our good friend Be^Rt, or bert. A nice expression for calculating continuously compounded interest, where B is the initial debt value (base), e is Euler's number, r is the interest rate, and t is the amount of time that has passed. You would want r and t to have the same time units and B and r the same money units, of course.

But that's just the expression, it doesn't really explain why it's like that. The fact that there's a really weird constant in there (Euler's number) is also not exactly intuitive.

The basic idea, though, is simple. Imagine this was discrete instead of continuous. Then you'd write B(1+r)^t. So if you had a rate of 5% interest per unit of time, you'd have B(1.05)^t. If three units of time pass, this just translates to B(1.05)(1.05)(1.05), for example. This always works for discrete compounding interest and is just handy on its own for back of the envelope math.

You can think of continuously compounded interest as being derived from the discrete form, where it asks a question: what if the amount of time that has passed can be arbitrarily small before we compound? This is usually framed as writing the discrete expression using time deltas instead of time and "taking the limit", which is a calculus term for saying, "make the variable(s) go very small and see what happens".

Imagine if you had a rate of 5% per year but wanted to compound every month. You could of course just convert your time values into 1/12 of a year and go from there. Taking the limit to compound continuously says, "what if I just made the time interval smaller and smaller? Would I get a nice, tidy expression?"

It turns out that when you do that, you get our friend bert.