From Pennies to e

Start with $1 earning 100% interest for one year. Compound it once and you have $2. Compound twice and you get (1 + 1/2)² = $2.25. Compound 10 times and you reach about $2.59.

The more often you compound, the closer the total creeps to $2.71828. That ceiling is e, the natural base of continuous growth.

Written precisely, e = lim (1 + 1/n)ⁿ as n grows without bound. That single limit pins down the value exactly.

e is irrational — like pi, its digits run on forever without repeating. It is also transcendental: no polynomial equation with whole-number coefficients can have e as a solution.