"What If We Compounded Interest Even More Frequently?"
Deposit 10,000 yen at 100% annual interest and after one year you have 20,000 yen. But what if interest is compounded every six months at 50%? Then 10,000 x 1.5 x 1.5 = 22,500 yen - 2,500 yen more than annual compounding. Monthly (12 times)? 10,000 x (1 + 1/12)^12 = roughly 26,130 yen. Daily (365 times)? 10,000 x (1 + 1/365)^365 = roughly 27,146 yen.
The more frequently you compound, the larger the final amount. So if you compound infinitely often, does the amount grow without limit? In 1683, Swiss mathematician Jacob Bernoulli set out to answer exactly this question.
The Answer Is "It Does Not Grow Without Limit" - and e Appears
Bernoulli's calculation revealed something remarkable: even as the compounding frequency approaches infinity, the amount converges to a finite value. Taking the limit of 10,000 x (1 + 1/n)^n as n approaches infinity yields approximately 27,183 yen - about 2.7183 times the original deposit. That number, 2.71828..., would later become known as Euler's number e, one of the most important constants in mathematics.
10,000 Yen at 100% Annual Interest, One Year Later
| Compounding interval | Calculation | Amount after one year | Gain over the row above |
|---|---|---|---|
| Once a year | 10,000 x 2 | 20,000 yen | - |
| Every six months | 10,000 x (1 + 1/2)^2 | 22,500 yen | +2,500 yen |
| Every three months | 10,000 x (1 + 1/4)^4 | 24,414 yen | +1,914 yen |
| Monthly | 10,000 x (1 + 1/12)^12 | 26,130 yen | +1,716 yen |
| Daily | 10,000 x (1 + 1/365)^365 | 27,146 yen | +1,016 yen |
| Infinitely often | 10,000 x e | approximately 27,183 yen | +37 yen |
Shortening the interval does add money, but each step down the table adds less than the one before. Going from daily all the way to infinitely often is worth only about 37 yen. Divide that ceiling of 27,183 yen by the original 10,000 yen and you get 2.71828... - that is what e really is.
Just as pi (3.14159...) arises from the geometry of circles, e (2.71828...) arises from the mathematics of compound interest. The e you encounter in textbooks was actually discovered through a thoroughly practical question: "What happens when you make interest payments infinitely fine?"
e Is Hiding Everywhere, Not Just in Finance
Born from compound interest, e has escaped the world of finance and appears throughout nature. Radioactive decay rates, bacterial growth curves, the shape of suspension-bridge cables, even the attenuation of your smartphone's radio signal - all are described by equations containing e. The fact that a number discovered through a money problem became a tool for describing the laws of the universe shows that compound interest is not merely a financial technique but a fundamental principle of nature.