What is e (Euler's number)?

e ≈ 2.718281828459045 is a mathematical constant: the limit that $1 invested at 100% interest approaches as the interest is compounded more and more often, and equivalently the sum 1 + 1 + 1/2 + 1/6 + 1/24 + … of reciprocal factorials. It is irrational and transcendental, so it is never a fraction and never a root of a whole-number polynomial.

Why does compound interest stop at e?

$1 at 100% annual interest compounded n times a year grows to (1 + 1/n)^n. Compounding more often helps less and less: the value rises toward e ≈ 2.71828 and the remaining gap shrinks like e/(2n), so even compounding every second only reaches 2.7182818… It approaches e from below and never reaches it at any finite n; the limit (continuous compounding) is exactly e.

Is e the same as 2.718?

2.718 is e rounded to three decimals. The full value 2.718281828459045… never terminates and never repeats, because e is irrational (proved by Euler, 1737).

Artificial Wasteland · search-demand stratum

What is e?
The number that won't be rushed.

It is about 2.718. But where does it come from, and why does it turn up in compound interest, in calculus, and in the odds of a coat-check disaster — places that seem to have nothing to do with each other? Put your hands on it.

e = 2.718281828459045235360287471352662497757…

1 · where it was found — 1683

A dollar, and an impossibly generous bank

Jacob Bernoulli asked a strange question. Put $1 in an account paying 100% interest per year. Paid once, you end with $2. But what if it's paid in installments — half-rate twice a year, each adding to the last? More often is more money… but how much more? Drag the dial.

Compounding frequency — once a year (n = 1)

$2.00000000

Formula: (1 + 1/n)ⁿ with n = 1.

The dashed ceiling is e. Notice: each step up the dial buys less than the last. The remaining gap to e shrinks like e / 2n — so it never closes. Even compounding every second (n ≈ 31.5 million) reaches only 2.7182818, still short of e in the far decimals.

2 · the express train

The same number, reached in seven steps

Compounding crawls toward e. There is a faster road: add up the reciprocal factorials, 1 + 1 + ½ + ⅙ + 1/24 + … Each term is a whole-number fraction, so every partial sum is exact — and it rockets to e. Add terms one at a time.

partial sum after 1 term · error from e: 1.718

krunning sum (exact)decimal

By just the 8th term the series is closer to e than compounding 8,760 times a year ever gets. Factorials in the denominator crush the error: it falls faster than any geometric rate.

3 · why this number and not 2, or 3

The one curve that is its own shadow

Pick a base b and draw y = b^x. Its steepness at the crossing point x = 0 is some number — the slope of the little arrow below. For most bases the curve and its own slope disagree. There is exactly one base where the height of the curve equals its slope everywhere — where the function is its own rate of change. Hunt for it.

base b — 2.000

slope at x=0 = 0.693147

When the slope reads exactly 1.000000, the curve has caught its own shadow. That happens at b = e.

4 · where e hides with no money in sight

Nobody gets their own hat

n people check their hats. The attendant panics and returns them at random, one per person. What's the chance that not a single person gets their own hat back? You'd guess it plunges toward zero as the crowd grows. It doesn't — it homes in on 1/e ≈ 0.3679, and stops. Shuffle the cloakroom.

crowd size — 7 people

P(nobody) = 0.367857…

Exact probability D₇/7! = 1854/5040. Rose tiles = someone got their own hat. The exact value sits a hair from 1/e and barely moves as you add people.

The check — every number here is recomputed live

Compounding. The balance is (1+1/n)ⁿ, evaluated as exp(n·ln(1+1/n)) for accuracy near the limit. It rises monotonically and stays strictly below e at every finite n; the residual gap divided by e/2n → 1 (checked at n = 365; 8,760; 525,600).
The series is summed as exact fractions with BigInt (denominator = k!), shown above as 5/2, 16/6, 65/24, … — never as rounded decimals forced into agreement.
Derangements use the exact recurrence Dₙ = n·Dₙ₋₁ + (−1)ⁿ → 1, 0, 1, 2, 9, 44, 265, 1854, 14833 (OEIS A000166); the displayed probability is the exact rational Dₙ/n!. The live 1000-shuffle simulation is a genuine random trial — it lands near the exact value, not on it.
The own-derivative base. The slope of bˣ at 0 is ln(b), equal to 1 only at b = e — recomputed as you drag.

The offline verifier research/eulers-number/verify.mjs runs all of these in exact arithmetic and must pass before this page ships. Open the page source: the same formulas run in front of you.

What is e?The number that won't be rushed.