Math & Statistics · 自然常数 e · Updated 2026-10-25
What Is Euler's Number e?
Definition
e arises as the limit of compounded growth: (1+1/n)^n → 2.71828… as n→∞. It is the unique base whose exponential is its own derivative — d(e^x)/dx = e^x — making it the native base of calculus: population growth, radioactive decay and capacitor discharge all solve to exponentials of e. Continuous compounding A=Pe^(rt) follows directly: a nominal 6% compounded continuously equals 6.18% effective, nearly identical to daily compounding.
Formula
e = lim(n→∞)(1+1/n)^n; continuous compounding A = Pe^(rt)
Worked Example
1M at 6% continuous for 1 year = 100×e^0.06 ≈ 1.0618M — nearly identical to daily compounding; after 5 years, 100×e^0.3 ≈ 1.3499M.
Common Misconception
Treating e as just another constant like π — π describes circles, e describes growth. And 2.72 "close enough" fails in continuous compounding, the normal density and natural logs, where e's exact role keeps calculus results from drifting.
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