What is e?
Suppose a bank pays 100% interest a year on 1. Added once a year, you end with 2. Added twice, each half-year multiplies by 1.5, giving 1.5² = 2.25. Added n times, you get (1 + 1/n)^n. As n grows the result keeps rising, but ever more slowly, toward a limit: e ≈ 2.718281828.
Computed as e^(n·ln(1 + 1/n)). Computing 1 + 1/n directly fails for very large n: a computer rounds 1 + 1/10¹⁶ to exactly 1, and 1 to any power is 1.
Periodic and continuous compounding
The same idea holds for any rate r: adding r/n interest n times a year approaches e^r per year. For 1,000 at 5% for 10 years:
The Compound Interest Calculator works this out for your own savings, with regular deposits.
Why e appears in growth
Exponential growth means the rate of change is proportional to the amount: dN/dt = kN. The function whose rate of change equals itself is e^t, so the solution is N(t) = N₀e^(kt). Positive k grows; negative k decays.
The natural logarithm ln undoes e: if e^x = y then ln y = x. That is why a doubling time is ln 2 / k (solve e^(kT) = 2), a half-life is ln 2 / |k|, and the time to reach a target is ln(target/N₀) / k.
A percentage per period and a rate constant differ: growing 5% each hour multiplies by 1.05 an hour, which is k = ln 1.05 ≈ 0.0488. Logarithmic growth, such as a + b·ln t, is a different function again: it keeps slowing and never accelerates.
Further reading
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