A Worked Example

Put the idea to work. Deposit $1,000 at 5% annual interest. Compounded once a year, the balance after 10 years is $1,000 × 1.05¹⁰ ≈ $1,629. Compounded continuously, it is $1,000 × e^(0.5) ≈ $1,649.

That roughly $20 gap is exactly what continuous compounding adds — the clearest way to see e's meaning in dollars.

The number e appears in ideal continuous-compounding models, radioactive decay, cooling, population models, and many other exponential processes. Real banks often compound daily, monthly, or by another schedule, so e is the limiting model rather than a guarantee about your account. Whenever a rate of change is proportional to the current amount, e is likely nearby.