Why is e Everywhere?

Almost every student meets the number $e \approx 2.71828$ in a calculus class, often introduced as “a number that just happens to make derivatives of exponentials clean.” That framing makes $e$ sound like a technical convenience. It isn’t. $e$ keeps showing up in problems that have nothing obvious to do with each other: a savings account at a bank, the decay of a radioactive atom, the probability of misaddressing every single envelope at a wedding. Something deep is going on.

This article is an attempt to explain, in plain language, why a single irrational number is the right answer to so many unrelated questions.

The compound-interest origin story

The cleanest way to meet $e$ is through Jacob Bernoulli’s original 1683 puzzle about compound interest.

Suppose a bank offers 100% annual interest. If it compounds once per year, $1 becomes$ 2. If it compounds twice per year at 50% each period, $1 becomes$ 1 × 1.5 × 1.5 = $2.25. Four times a year at 25% each:$ 1 × 1.25⁴ ≈ $2.441. Monthly compounding gets you to about$ 2.613. Daily: about $2.7146.

Bernoulli asked: what if we compound continuously — an infinite number of times per second, each time at an infinitesimally small rate? The answer isn’t infinity. It converges to a specific number:

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

That limit is the definition of $e$ . Every subsequent appearance is, in some sense, a restatement of this same limit in different clothing.

Growth proportional to size

The reason $e$ is so universal becomes clearer once you notice what compound interest really models: a quantity that grows at a rate proportional to its current size.

That single description fits an enormous number of natural processes:

A bacterial colony doubles in size by having each existing cell split.
A savings account earns interest on the money it already contains.
A population reproduces roughly in proportion to its current headcount.
A radioactive sample decays with each atom independently having the same chance of breaking down per unit time.

In every case the governing equation is the same:

$\frac{dN}{dt} = k \cdot N$

And the solution to that equation — the function whose derivative is proportional to itself — is

$N(t) = N_0 \cdot e^{kt}$

Any time you see exponential growth or exponential decay, you’re looking at $e$ in disguise. Change of variable can turn $e^{kt}$ into $2^t$ or $10^t$ , but $e$ is the “natural” base because it makes the derivative clean: $\frac{d}{dx} e^x = e^x$ . Nothing else has that property.

Where else e shows up

Once you know to look for it, $e$ surfaces in surprising places.

The normal distribution. The bell curve’s density function contains $e^{-x^2/2}$ . The Gaussian is the limit of many independent random additions (the Central Limit Theorem), and that limit produces $e$ because the probability structure of averaging is exponential in the log-likelihood.

Derangements. If $n$ people randomly pick coats from a checkroom, what’s the probability nobody gets their own? For large $n$ the answer is almost exactly $1/e \approx 36.8\%$ . The derivation counts permutations with no fixed points; the series that falls out is the same Taylor series that defines $e^{-1}$ .

The secretary problem. If you must hire someone after interviewing candidates one by one (no going back), the optimal strategy is to reject the first $n/e$ candidates and then hire the next one who beats all those you’ve seen. The probability of landing the best candidate is also $1/e$ .

Euler’s identity. The most famous equation in mathematics:

$e^{i\pi} + 1 = 0$

links $e$ to $\pi$ , $i$ , $1$ , and $0$ . That identity falls out of extending $e^x$ to complex numbers via its Taylor series, which shows that $e^{ix} = \cos x + i \sin x$ . The exponential and the circle are secretly the same function.

The Taylor series perspective

The deepest answer to “why is $e$ everywhere?” is probably this: $e^x$ has an unusually clean Taylor series.

$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$

That series converges for every real or complex $x$ , and each term has a clear meaning: the $n$ -th term is how much the function has grown after $n$ stages of compounded change. Many problems in probability, combinatorics, and analysis naturally produce series of the form $\sum x^n/n!$ , and whenever that happens you’re implicitly computing $e^x$ .

This is also why $e$ is the base of the natural logarithm. The integral $\int_1^x 1/t \, dt$ — the area under $1/t$ — defines a function whose inverse is $e^x$ . Using any other base introduces an awkward multiplicative constant.

What to remember

If you want one mental model for $e$ : it’s the number you get whenever something grows in proportion to its own size, and it’s the only base that makes calculus clean. Every other appearance — derangements, the normal distribution, Euler’s identity — is ultimately a restatement of that same idea filtered through different notation.

The universe doesn’t care about base 10 or base 2. Those are artifacts of having ten fingers and of binary hardware. When a process evolves purely by its own internal logic, it evolves in base $e$ .

Frequently asked

Is e more fundamental than π?

Both are equally fundamental — they sit on opposite sides of Euler's identity. π captures rotation and circles; e captures growth and change. Whenever something scales in proportion to its own size, e appears.

Why exactly 2.71828…?

Because e is defined as the limit of (1 + 1/n)^n as n grows. Numerically that limit converges to 2.71828182845…, and the choice of base makes the derivative of e^x equal to itself — a property no other base has.

The compound-interest origin story

Growth proportional to size

Where else e shows up

The Taylor series perspective

What to remember

Frequently asked

Continue reading

Euler's Identity

The Gaussian Distribution

Leonhard Euler

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