Christoffer De Geer March 20, 2026

What is e (Euler's Number)?

e = lim(1 + 1/n)ⁿ ≈ 2.71828…

e ≈ 2.71828182845904523536. Irrational and transcendental.

e is the unique number where the function eˣ is its own derivative. Start with any amount and let it grow continuously at 100% per year. After exactly one year you have e times what you started with. No other base shares this self-referential property.

The limit definition: (1 + 1/n)ⁿ → e

As n grows, the sequence approaches e from below, converging to 2.71828182845904…

The limit definition: (1 + 1/n)ⁿ → e

Table showing (1+1/n)^n converging to e
n	(1 + 1/n)ⁿ	distance to e
1	2.000000	0.71828
10	2.593742	0.12454
100	2.704814	0.01347
1 000	2.716924	0.00136
1 000 000	2.718281	0.0000014
∞	2.71828…	0

The compound interest interpretation: if a bank pays 100% annual interest but compounds it n times per year, your balance grows by (1 + 1/n)ⁿ. Compounding monthly gives 2.613. Compounding every second gives 2.718. Compounding continuously gives exactly e.

e^x: the only function that is its own derivative

At x=1, the height of the curve is e ≈ 2.718 and the slope of the tangent is also e. No other base b^x has this property.

Jacob Bernoulli discovered e in 1683 while studying compound interest. Euler named it e in 1731. It is irrational (Euler, 1737) and transcendental (Hermite, 1873). Its decimal expansion 2.71828182845904523536… never repeats.

Euler's Identity → Natural Log of 2 → Taylor Series → Stirling's Approximation → Continued Fractions → The Major System →

Compound interest converges to e as compounding increases

Starting with $1 at 100% annual interest: compounding monthly gives $2.613, daily $2.714, every second $2.718. The limit as n→∞ is exactly e.

Key facts about Euler's Number e

e (Euler's number) is approximately 2.71828182845904523536. It is the unique number where the function e^x equals its own derivative at every point. Jacob Bernoulli discovered it in 1683 studying compound interest. Leonhard Euler named it e around 1731. e is irrational (Euler, 1737) and transcendental (Hermite, 1873). It appears in continuous growth and decay, natural logarithms, the normal distribution, compound interest, radioactive decay, and Euler's identity e^(i*pi) + 1 = 0.

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#memoryAll memory gamesHold a sequence, grid, or list in working memory and reproduce it under pressure.#numbersAll numbers gamesArithmetic, ratios, units, and number sense in short timed rounds.#sequenceAll sequence gamesReproduce or extend an ordered string of items in the right order.