Euler's Number Formula

Q: What is the Euler's Number formula?

Euler's number $e \approx 2.71828$ is the unique base for which the exponential function $e^x$ is its own derivative — the natural base for growth and decay.

Q: What do the symbols mean in the Euler's Number formula?

$e$ denotes Euler's number. $e^x$ or $\exp(x)$ denotes the natural exponential function.

Q: What do students get wrong about Euler's Number?

$e$ is irrational and transcendental—its digits never repeat or end.

The Formula

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

When to use: The 'natural' base for growth—what you get from continuous compounding.

Quick Example

e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n Compound interest continuously: \1 becomes e$ after 1 year at 100%.

Notation

e denotes Euler's number. e^x or \exp(x) denotes the natural exponential function.

What This Formula Means

Euler's number e \approx 2.71828 is the unique base for which the exponential function e^x is its own derivative — the natural base for growth and decay.

The 'natural' base for growth—what you get from continuous compounding.

Formal View

e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^{\!n} = \sum_{k=0}^{\infty}\frac{1}{k!} \approx 2.71828

Worked Examples

Example 1

easy

Evaluate \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n and state what this limit defines.

Solution

1
Substitute increasing values of n: for n=10, \left(1+\frac{1}{10}\right)^{10} \approx 2.5937; for n=100, \approx 2.7048; for n=1000, \approx 2.7169.
2
Observe that as n \to \infty the expression approaches a fixed value that does not grow without bound.
3
By definition, \lim_{n \to \infty}\left(1+\frac{1}{n}\right)^n = e \approx 2.71828\ldots, Euler's number.

Answer

e \approx 2.71828

Euler's number e is defined as this limit. It is irrational and transcendental, appearing naturally in continuous growth, calculus, and complex analysis.

Example 2

hard

Show that the derivative of f(x) = e^x is itself, i.e., f'(x) = e^x, using the limit definition of the derivative.

Common Mistakes

Thinking e is a variable — e \approx 2.71828 is a fixed constant, like \pi
Rounding e to 3 in calculations — e \approx 2.718, and the difference matters in exponential growth over time
Confusing e^x with xe — e^2 \approx 7.389, while 2e \approx 5.436; exponentiation is very different from multiplication

Why This Formula Matters

Appears naturally in growth, decay, probability, and calculus.

Frequently Asked Questions

What is the Euler's Number formula?

Euler's number e \approx 2.71828 is the unique base for which the exponential function e^x is its own derivative — the natural base for growth and decay.