What is e (Euler's Number)?
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.
As n grows, the sequence approaches e from below, converging to 2.71828182845904…
| 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.
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.
Starting with $1 at 100% annual interest: compounding monthly gives $2.613, daily $2.714, every second $2.718. The limit as n to ∞ is exactly 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.
Euler's Number e is irrational. Its decimal expansion never ends and never repeats. The digits shown below are verified against the taylor series.
FAQ
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Who discovered e and when?
Jacob Bernoulli first encountered the constant in 1683 while studying compound interest. Leonhard Euler began using the letter e around 1731 and was the first to publish systematic results about its properties. -
Why is the constant called e?
Euler first used the letter e in an unpublished paper around 1728. Whether it stands for Euler or exponential is debated. The designation stuck through the influence of his extensive published work. -
When was e proven to be transcendental?
Charles Hermite proved e is transcendental in 1873, making it the first number ever proven to be transcendental. This inspired Lindemann to prove \u03C0 transcendental a decade later using similar methods. -
Where does e appear in finance?
Continuous compounding uses e: if you invest principal P at annual rate r for t years, the value is P times e^(rt). Jacob Bernoulli found the constant precisely this way - studying what happens when interest is compounded infinitely many times per year. -
What is the natural logarithm, and why is it natural?
The natural logarithm (ln) is the logarithm base e. It is called natural because it is the anti-derivative of 1/x, the simplest non-trivial function. The term was introduced by Nicholas Mercator in his 1668 work Logarithmotechnia.
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