Maths Solver$e$ is approximately $2.71828$. It is the base of the natural logarithm, it appears all over calculus, statistics, and physics, and most explanations of where it comes from involve at least a paragraph about limits or derivatives. That is fine in a calculus course, but it makes $e$ feel like a piece of advanced mathematics that you cannot hope to understand until you can already do calculus.
You can. Compound interest is enough.
The compound-interest setup
Suppose I lend you $\pounds 1$ for one year at $100\%$ interest. At the end of the year, you owe me $\pounds 2$. Easy. Now suppose I compound the interest twice a year — that is, I charge you $50\%$ at the half-year mark and another $50\%$ at the year-end. Half way through the year you owe $\pounds 1.50$, and the second $50\%$ is charged on the larger amount. So at year-end you owe $\pounds 1.50 \times 1.50 = \pounds 2.25$.
Compound it four times a year ($25\%$ per quarter) and the total becomes $\pounds 1 \times (1.25)^4 = \pounds 2.44$. Compound it twelve times a year (monthly) and you get $\pounds 1 \times (1 + \tfrac{1}{12})^{12} \approx \pounds 2.61$. The more often you compound, the bigger the year-end total — because the interest you have already accrued is itself accruing interest.
The natural question: what if you compound infinitely often? Does the year-end total go to infinity? It feels like it should. The interest keeps accruing on the interest, and there is no obvious cap.
| Compounding | Value of $(1 + 1/n)^n$ |
|---|---|
| Annually ($n = 1$) | 2.0000 |
| Twice a year ($n = 2$) | 2.2500 |
| Quarterly ($n = 4$) | 2.4414 |
| Monthly ($n = 12$) | 2.6130 |
| Daily ($n = 365$) | 2.7146 |
| Hourly ($n = 8{,}760$) | 2.7181 |
| Every second ($n \approx 31\,$M) | 2.7183 |
| Continuously ($n \to \infty$) | $e \approx 2.71828\ldots$ |
The total does not go to infinity. It approaches a fixed number, just under $2.72$. That number is $e$. It is, by definition, what you get when you compound a $100\%$ annual interest rate as often as physically possible.
$e$ is irrational, and a bit weirder than that
The decimal expansion of $e$ never terminates and never repeats. That makes $e$ irrational, in the same family as $\pi$ and $\sqrt{2}$. But $e$ is also transcendental, which is a stronger condition: it is not the root of any polynomial with integer coefficients. (For contrast, $\sqrt{2}$ is irrational but is the root of $x^2 - 2 = 0$, so it is not transcendental.)
The fact that $e$ is irrational was proved by Euler in 1737; the fact that it is transcendental was proved by Hermite in 1873. Both proofs are well outside school maths, but neither fact requires you to be able to do them in order to use $e$.
Why $e$ shows up everywhere
The reason $e$ is so ubiquitous is not actually because of compound interest, even though that is the cleanest place to meet it. The deeper reason is that the function $f(x) = e^x$ has a very special property in calculus: it is its own derivative. That is to say, the slope of the graph of $e^x$ at any point equals the value of $e^x$ at that point. No other function, except multiples of $e^x$, has this property.
This is more useful than it sounds. Most growth processes in physics and biology — population growth, radioactive decay, capacitor charging, Newton’s law of cooling — are described by an equation of the form “the rate of change is proportional to the amount currently present.” The unique function that satisfies that description is some constant times $e^{kt}$, for some rate constant $k$. So $e$ is not just the “continuous compounding constant” in finance; it is the natural mathematical representation of any process where the rate of change scales with the current value.
That class of processes is large enough that $e$ gets pulled into nearly every quantitative subject as a result. It is in the exponential distribution in statistics, in the heat equation, in the black-body radiation spectrum, in the formula for entropy. None of these were on anyone’s mind when $e$ was first defined as a compound-interest limit, but they all turn out to involve the same object.
The other definition you will see
Once you know calculus, you are likely to meet $e$ defined a different way: as the unique number $a$ such that the function $a^x$ has slope $1$ at $x = 0$. This is equivalent to the compound-interest definition, but it is harder to motivate without already having calculus — which is the chicken-and-egg problem that makes “what is $e$?” an awkward question to answer to a beginner.
There is also a series definition:
$$e = \sum_{n=0}^{\infty} \frac{1}{n!} = 1 + 1 + \tfrac{1}{2} + \tfrac{1}{6} + \tfrac{1}{24} + \cdots$$
This converges very fast — the first ten terms get you eight correct decimal places — and it is how $e$ is actually calculated to high precision in practice. It also looks like it has nothing to do with compound interest, but the equivalence between this series and the compound-interest limit is one of the standard results in introductory calculus.
How to think about $e$ before you have calculus
A small mental model that has worked for many of my students: $e$ is the answer to the question “what is the natural rate of exponential growth?” If something doubles in some fixed amount of time at the simplest possible compounding regime, the cleanest description of that doubling is in terms of $e^{kt}$, where $k$ is whatever rate makes the doubling happen at the right moment. The $e$ is doing the work of being “the natural exponential base,” in the same way $\pi$ does the work of being “the natural constant for circular things.”
Both numbers earn their universal status the same way: they each turn out to be the simplest description of a phenomenon that comes up in dozens of different areas. $\pi$ for things involving rotation or periodicity, $e$ for things involving growth or decay.
Things that are not $e$
A few common misconceptions worth flagging:
- $e$ is not Euler’s number in the sense of being his single most important contribution. Euler made dozens of huge contributions to mathematics; $e$ is named after him because he was the first to use the symbol systematically (around 1731), not because he discovered it.
- $e$ is not equal to $2.7$, or $2.72$, or even $2.71828$. All of those are truncations. The true value is irrational.
- $e^{i\pi} = -1$ (Euler’s identity) is real but not magic. It comes from the fact that $e^{ix}$ rotates points around the unit circle in the complex plane, and rotating $1$ by $\pi$ radians takes you to $-1$. The identity is striking; it is not mysterious once you have seen the geometry.
Where to go next
If you want to see $e$ used in practice, the compound-interest article on this site walks through the continuous-compounding formula $A = Pe^{rt}$, which is exactly the limit we computed at the top of this post. The derivative guide is where the “$e^x$ is its own derivative” property gets proper treatment.
For a more visual introduction, 3Blue1Brown has a video on YouTube called “What’s so special about Euler’s number?” that builds the intuition with animations in a way text cannot. It is worth ten minutes if you are still mystified.