The Natural Constant $e$

The constant $e \approx 2.71828$ emerges naturally from growth and decay, logarithms, and calculus.

From Compound Interest to $e$

Consider $1$ unit of principal with 100% annual interest.

• Compounded yearly: $(1+1)^1 = 2$
• Quarterly: $\left(1+\dfrac{1}{4}\right)^4 \approx 2.4414$
• Monthly: $\left(1+\dfrac{1}{12}\right)^{12} \approx 2.6130$
• Continuously: $\lim_{n\to\infty}\left(1+\dfrac{1}{n}\right)^n = e$

Limit definition of $e$

$$e = \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$$

Key Properties

• $e$ is irrational (and transcendental).
• Natural logarithm: $\ln e = 1$, $\ln 1 = 0$.
• Derivative: $(e^x)' = e^x$; $(\ln x)' = \dfrac{1}{x}$ for $x>0$.
• Exponential $e^x$ always positive, crosses $(0,1)$, increases on $\mathbb{R}$.

Common Applications

• Continuous growth/decay models: $P(t) = P_0 e^{kt}$.
• Limit computations and series expansions.
• Change-of-base for logarithms: $\log_a x = \dfrac{\ln x}{\ln a}$.

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练习题

练习 1

Evaluate $\displaystyle \lim_{n\to\infty}\left(1+\dfrac{2}{n}\right)^n$.

参考答案

Rewrite $\left(1+\dfrac{2}{n}\right)^n = \left[\left(1+\dfrac{2}{n}\right)^{\tfrac{n}{2}}\right]^2 \to e^{2}$.

练习 2

If $P(t) = 100 e^{0.05t}$, find $P'(t)$ and interpret.

参考答案

$P'(t) = 100 \cdot 0.05 \, e^{0.05t} = 5 e^{0.05t}$. The amount grows at a rate proportional to its current value (5% per time unit).

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总结

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$e$	数学符号	e	自然常数，约 $2.71828$
$\ln x$	数学符号	natural log of x	以 $e$ 为底的对数
$e^x$	数学符号	e to the x	以 $e$ 为底的指数函数
$P(t)$	数学符号	P of t	随时间变化的数量

中英对照

中文术语	英文术语	音标	说明
自然常数	natural constant e	/ˈnætʃrəl ˈkɒnstənt iː/	由连续复利与极限定义的常数
自然对数	natural logarithm	/ˈnætʃrəl ˈlɒɡərɪðəm/	以 $e$ 为底的对数
连续复利	continuous compounding	/kənˈtɪnjʊəs kəmˈpaʊndɪŋ/	利息连续计入本金的模型
指数增长	exponential growth	/ˌɛkspəˈnɛnʃəl ɡrəʊθ/	比例与当前量成正比的增长

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