e = 2.718

e is the maximum possible result of continuously compounding 100% over one time period.

All of these things are equal:

\(e^{1/x} = 10\) \(10^x = e\) \(x = log_{10}(e)\) \(x = [log_e(10)]^{-1}\) \(log_e(10) \approx 2.3\)

Formula

Take yearly compounding, say 100% growth per period.

Given $1 you would have $2 after one period.

Split that one period into two periods and compound by 50% twice and you get 1. 1 + (1 * .5) = 1.5 for the first half and 2. 1.5 + (1.5 * .5) = 2.25 - Clearly better

100% annual rate, compounded four times per period: 1. 1 + (1 * .25) = 1.25 2. 1.25 + (1.25 * .25) = 1.5625 3. 1.5625 + (1.5625 * .25) = 1.953125 4. 1.953125 + (1.953125 * .25) = 2.44

It goes up more, but not by the same amount. The more you go up, the more it begins to converge around 2.71. Here’s a table:

Applicability for other growth rates

Often, e is given as the formula: \(e^{rt}\)

If we wanted to denote 200% growth over a period of 5 years, it would be: \(e^{2 \times 5} = e^{10}\)