Euler's Number¶

Definition¶

Euler's number $\mathrm{e}$ is defined by the standard sequence limit

\mathrm{e}=\lim\limits_{n\to\infty}\left(1+\frac{1}{n}\right)^n,

which is an irrational number approximately equal to $2.71828182846\text{.}$

Interpretation¶

Euler's number $\mathrm{e}\approx2.718$ arises naturally from processes involving continuous growth. If an amount grows by $100\%$ per period and the growth is credited $n$ times per period, each time by a fraction $\tfrac{1}{n}$ of the current amount, the amount is multiplied by $\left(1+\tfrac{1}{n}\right)^n$ ; the definition lets $n\to\infty$ $\infty$ .

Two panels: the graph of e to the x with its tangent y = x + 1 at 0 and the value e at x = 1, and the sequence (1 + 1/n) to the n increasing towards e.

The right panel shows the defining sequence: $\left(1+\tfrac{1}{n}\right)^n$ equals $2$ for $n=1$ $1$ , $2.25$ for $n=2$ and about $2.594$ for $n=10$ $10$ . The terms increase towards $\mathrm{e}$ , but slowly: crediting more often helps less and less, and the result stays below $\mathrm{e}$ . The left panel looks ahead to the curve $y=\mathrm{e}^x$ $e^{x}$ , which passes through $(0,1)$ with slope $1$ , touching the line $y=x+1$ there, and reaches the height $\mathrm{e}$ at $x=1$ $1$ .

Increase $n$ and watch $\left(1+\frac1n\right)^n$ creep up towards $\mathrm{e}$ , slowly. Switch to $\left(1+\frac rn\right)^n$ to see the whole curve approach $\mathrm{e}^r$ .

A common slip is to take the limit of the base first: $1+\tfrac{1}{n}\to1$ and $1^n=1$ $1$ , which suggests the limit $1$ . This is wrong because the base and the exponent change together. The limit laws cover sums, products and quotients of convergent sequences, so they give $\left(1+\tfrac{1}{n}\right)^k\to1$ for a fixed exponent $k$ , but they say nothing about a power whose exponent grows with $n$ . Later, $\mathrm{e}$ becomes the base of the exponential function and of the natural logarithm, and it appears in probability, for example in the Poisson distribution.

Key Results¶

Euler’s number is irrational; Math I: Calculus uses this fact without proving it.
The natural exponential function is defined using $\mathrm{e}\text{.}$
In Math I: Calculus, this limit is used as the definition of $\mathrm{e}\text{;}$ proving that the defining sequence converges is optional.

Examples¶

$\displaystyle \lim_{x\to 0^+}\left(1+x\right)^{1/x}=\mathrm{e}\text{.}$
$\displaystyle\lim\limits_{n\to\infty}\left(1+\frac{r}{n}\right)^n=\mathrm{e}^r\text{.}$

Prerequisites¶

Limit of a Sequence

Sources¶

Math I: Calculus · Lecture 3

Aliases: e, Napier's constant, base of the natural logarithm, base of the natural exponential Symbols: infinity

Last updated: 20 October 2026 03:02