If a bank pays 100% interest a year on one coin and splits that interest into n equal payments, each one added to the balance, the coin grows to (1 + 1/n)n by the end of the year. Paid once, you get 2. Paid twice, 2.25. Quarterly, 2.4414. Monthly, 2.6130. Daily, 2.7146. Hourly, 2.7181. The balance keeps rising, but it never crosses e ≈ 2.71828, the limit of (1 + 1/n)n as n grows. That is how compound interest leads to the number e, and Jacob Bernoulli studied exactly this puzzle in 1683.
Jacob Bernoulli’s compound interest question
Jacob Bernoulli was born in Basel on 6 January 1655 and taught mechanics at the University of Basel from 1683. That same year, according to MacTutor’s history of e, he took up the problem of compound interest. If interest is added more and more often, approaching continuous compounding, what happens to (1 + 1/n)n as n tends to infinity?
Using the binomial theorem, he showed that the limit had to lie between 2 and 3, which MacTutor suggests could be seen as the first approximation of e. It adds that if we accept this as a definition of e, it is the first time a number was defined by a limiting process. His solution appeared in print in the journal Acta Eruditorum in 1690. He did not connect his number with logarithms, even though a table of natural logarithms printed in 1618, in an appendix to Napier’s work, already rested on it without anyone noticing.
The ladder of balances
Start with 1 coin at 100% a year. Paid twice, the first half-payment brings it to 1.5, and the second half-payment is 50% of 1.5, giving 1.5 × 1.5 = 2.25. The extra 0.25 is interest on interest. More payments mean more interest on interest:
| Paid | n | Balance after one year |
|---|---|---|
| yearly | 1 | 2 |
| twice a year | 2 | 2.25 |
| quarterly | 4 | 2.4414 |
| monthly | 12 | 2.6130 |
| weekly | 52 | 2.6926 |
| daily | 365 | 2.7146 |
| hourly | 8,760 | 2.7181 |
(Balances rounded to four decimals.) The video climbs this ladder rung by rung, and the gains shrink fast. Splitting into two payments added 0.25; going from daily to hourly adds only 0.0035. Paid every minute (n = 525,600), the coin reaches 2.718279…, and every second (n = 31,536,000) it reaches 2.718281785…. The limit is e = 2.718281828….
Why splitting the interest never makes you rich
Bernoulli’s bounds come from expanding the power with the binomial theorem:
(1 + 1/n)n = 1 + 1 + (1 − 1/n)/2 + (1 − 1/n)(1 − 2/n)/6 + …
The first two terms already make 2, so the balance is at least 2. The third term is less than 1/2, the fourth less than 1/6, the fifth less than 1/24, and so on. Those caps are no bigger than 1/2, 1/4, 1/8, …, which add up to 1. So the whole balance stays below 1 + 1 + 1 = 3, however finely you split the year.
As n grows, each bracket like (1 − 1/n) creeps toward 1, so the sum creeps toward 1 + 1 + 1/2 + 1/6 + 1/24 + …, which is e. The balance rises forever and never arrives.
Where e went next
The constant got its letter later. In 1690 Leibniz wrote to Huygens using the letter b for it. Leonhard Euler started writing e around 1727 or 1728, used it in a letter to Goldbach in 1731, and first put it in print in his Mechanica of 1736.
Bernoulli’s question is still the cleanest way to meet e. At a realistic rate r, the same limit gives (1 + r/n)n → er: 5% paid once turns 1 into 1.05, while 5% compounded continuously gives e0.05 ≈ 1.05127. The full story of the constant is in why e is 2.718, and it turns up even where no money changes hands, as in the puzzle of four letters in four envelopes.
However you split it, that number emerges.
