Euler Number (e) — Definition, Formula & Examples

The Euler number, written as

e

, is an irrational mathematical constant approximately equal to 2.71828. It serves as the base of the natural logarithm and appears throughout exponential growth, compound interest, and calculus.

The constant $e$ is defined as the limit $\displaystyle e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n$ . It is the unique positive real number such that the function $f(x) = e^x$ has a derivative equal to itself at every point.

Key Formula

e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n \approx 2.71828

Where:

$n$ = A positive integer that increases without bound

How It Works

When you use

e

as the base of an exponential function, you get

f(x) = e^x

, which has the special property that its rate of change equals its current value. This makes

e

the natural choice for modeling continuous growth or decay. On your calculator, the

e^x

button evaluates powers of

e

, and the

\ln

button computes the natural logarithm (log base

e

). Any exponential function

a^x

can be rewritten as

e^{x \ln a}

, which is why

e

is considered the most fundamental exponential base.

Worked Example

Problem: A bank offers 100% annual interest compounded continuously. If you invest $1, how much do you have after 1 year?

Set up the formula: Continuous compounding uses the formula

A = Pe^{rt}

, where

P = 1

r = 1

, and

t = 1

A = 1 \cdot e^{(1)(1)} = e^1

Evaluate: Since

e \approx 2.71828

, the investment grows to approximately $2.72.

A \approx 2.71828

Answer: After 1 year, $1 grows to approximately $2.72 under continuous compounding — this is exactly the value of

e

Why It Matters

The constant

e

is central to AP Calculus, where differentiation and integration of

e^x

are foundational skills. In finance,

e

powers the continuous compounding formula

A = Pe^{rt}

. Scientists and engineers rely on

e

to model radioactive decay, population growth, and electrical circuits.

Common Mistakes

Mistake: Confusing

e

with a variable or thinking its value changes.

Correction:

e

is a fixed constant like

\pi

. It always equals approximately 2.71828 and cannot be assigned a different value.