Resources:
- “e – Euler’s number,” Math is Fun, 2025. https://www.mathsisfun.com/numbers/e-eulers-number.html (accessed Feb. 01, 2026).
- C. to, “mathematical constant; limit of (1 + 1/n)^n as n approaches infinity; transcendental number approximately equal 2.718281828,” Wikipedia.org, Nov. 08, 2001. https://en.wikipedia.org/wiki/E_(mathematical_constant) (accessed Feb. 01, 2026).
What is Euler’s Number?
Euler’s number () was first discovered by Swiss mathematician in Jacob Bernoulli in1683 while studying compound interest. However, it is named after Leonhard Euler, who popularized the constant, discovered its connection to calculus, calculated its value to 23 decimal places, and used the letter 𝑒 for it in 1748.
- Discovery (1683): Jacob Bernoulli discovered the constant 𝑒 by investigating a question about compound interest, finding that the expression approaches a specific limit as 𝑛 increases.
- Definition & Naming (1748): Leonhard Euler defined 𝑒 as the base of the natural logarithm, showed it is an irrational number, and represented it as an infinite sum of inverse factorials ().
- Alternative Name: It is sometimes called Napier’s Constant in honor of John Napier, who invented logarithms, though Napier did not discover 𝑒 itself.
Definitions
$e$ is often defined as the limit \[ e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \] as $n$ approaches infinity.
It can also be expressed as the infinite sum \[ e = \sum_{n=0}^{\infty} \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots \]
Euler’s Identiy
The number $e$ is linked to the constant $\pi$ through Euler’s formula:
https://en.wikipedia.org/wiki/Euler%27s_formula
\[ e^{i\theta} = \cos \theta + i \sin \theta \]
for any real number $\theta$.
A famous special case, known as Euler’s identity, occurs when $\theta = \pi$:
https://en.wikipedia.org/wiki/Euler%27s_identity
\[
e^{i\pi} + 1 = 0
\]
This elegant equation connects five fundamental mathematical constants: $e$, $i$, $\pi$, $1$, and $0$.