Euler's Number

e is the number that makes calculus simple: d(eˣ)/dx = eˣ, and no other base has that property. It is defined, not measured, so the value stored here is the nearest double to the true constant. Jacob Bernoulli met it in 1683 asking what happens to compound interest as the compounding period shrinks — (1 + 1/n)ⁿ as n → ∞ — and found it converged to something between 2 and 3 without identifying what. Leonhard Euler named the constant e in a 1731 letter to Goldbach, computed it to 18 places, and proved it irrational in 1737; Charles Hermite proved it transcendental in 1873, nine years before Lindemann did the same for π by extending Hermite's method.

The naming has attracted more folklore than it deserves. Euler almost certainly did not name it after himself — he was using a, b, c and d for other things, and e was simply next, or possibly stood for "exponential". The constant turns up wherever a quantity grows or decays at a rate proportional to its own size: radioactive decay, RC circuit charging, bacterial growth, Newton's law of cooling, and the normal distribution. And it produces the identity most often voted the most beautiful in mathematics, e^{iπ} + 1 = 0, which ties together the five constants that matter most.