Why Is Euler’s Number So Special?

Euler’s number is approximately 2.718, shows up everywhere in nature, describes how things grow continuously, was discovered separately through compound interest calculations and calculus and limit equations, and turns out to be the same number every time even though there’s no reason it should be.

A Model Of Exponential Growth (3Blue1Brown)

That’s the weird part. If you invest money and your interest gets added to your account multiple times per year, the more often it compounds, the more money you end up with. But there’s a limit. Even if your bank compounds your interest infinitely many times per second, you can’t earn infinitely much money. You’ll hit a ceiling. That ceiling is e. If you start with one dollar at one hundred percent interest compounded continuously for one year, you end up with exactly e dollars, or about 2.718 dollars.

Now imagine you’re looking at a completely different problem. You’re studying calculus and you want to find the derivative of a function that’s growing. Most functions have a derivative that’s different from the function itself. The derivative of x squared is 2x. The derivative of x cubed is 3x squared. They’re all different. But there’s one special exponential function where the derivative equals the function itself. That function is e to the x power. The rate at which e to the x is growing at any point equals the value of e to the x at that point. It’s the only exponential function with this property. And the base of that exponential function is also e. The same 2.718 from the compound interest problem.

Nobody planned this. The two problems have nothing to do with each other. One comes from banking. One comes from calculus. They shouldn’t produce the same number. But they do.

Here’s another way e shows up. If you take a limit, the expression (1 + 1/n) raised to the n power, and let n get larger and larger, what do you get. Start with n equals one. You get 2. Now let n equal two. You get 2.25. Let n equal ten. You get 2.593. Let n equal one hundred. You get 2.704. Let n equal one thousand. You get 2.7169. Keep going forever and you approach 2.718. You get e again. The same number from a completely different direction.

This happens because e is what’s called a fixed point. It’s the number that certain processes naturally arrive at. If you’re looking at continuous growth, you end up at e. If you’re looking at how functions change, you end up at e. If you’re looking at how a limit converges, you end up at e. It’s not that someone picked e as a base and it happened to work. It’s that e is the base that anything growing continuously has to be described by.

This is why e shows up in population growth. If bacteria are dividing constantly, their population doesn’t grow in discrete jumps. It grows continuously. The rate of growth at any moment equals the size of the population at that moment. You double the population and the growth rate doubles too. This relationship, where the rate of change equals the thing changing, forces you to use e as your base. You don’t choose it. The system itself demands it.

Money works the same way. If you earn interest and that interest earns interest, your money grows continuously. The rate at which your money grows is proportional to how much money you have. More money means faster growth. This is the same relationship. It demands e. The only way to describe continuous compounding is with e as your base.

Even a cooling cup of coffee follows the same rule. Hot water cools down, but the rate at which it cools depends on how much hotter it is than the room. A very hot cup cools fast. A lukewarm cup cools slow. The rate of cooling equals the difference in temperature. This is the same relationship again. It forces e to appear.

None of these systems know about each other. A bacterium doesn’t know about your investment account. Your coffee doesn’t know about either one. But all of them produce e when you describe their continuous growth or decay mathematically. This is because they’re all following the same fundamental rule: the rate of change is proportional to the current amount.

The number e itself is irrational, which means it goes on forever without repeating. You can never write it exactly. You can only approximate it as 2.71828182845904523536 and so on, forever. You can calculate it using an infinite series. You add one plus one plus one over two plus one over two times three plus one over two times three times four, and keep adding smaller and smaller fractions forever. But even with a million terms, you still haven’t written out e completely. It exists as a limit. You approach it but never arrive.

This is fitting because e itself is about limits. It’s what you approach when you compound interest infinitely often. It’s what you approach when you take a limit to infinity. It’s what you get when you push continuous growth to its logical extreme. You never actually reach it in real life. A bank can’t compound your interest infinitely many times per year. Time can’t approach infinity. But the mathematics says there’s a destination out there, and that destination is e.

Leonhard Euler didn’t invent e. He discovered it while working on other problems. He saw it showing up in multiple places and recognized that it was fundamental. He started using the letter e to represent it. The letter e might stand for exponential. It might stand for Euler himself. Nobody knows for certain what he meant. But the letter stuck. Now whenever mathematicians see e, they think about growth and change and continuous processes.

What makes e very strange is that we still don’t have a simple geometric picture of why it appears everywhere. You can see why pi appears in circles. The ratio of a circle’s circumference to its diameter will always be pi. It’s visual and intuitive. But e appears in compound interest and bacterial growth and cooling cups of coffee, and there’s no simple picture that explains why all these things produce the same number. You have to work through the math to see it. You have to understand that anything growing at a rate proportional to itself will produce e. You have to recognize the pattern.

The practical result is that once you understand e and the natural logarithm (which is the inverse of e to the x), you can solve real world problems. You can calculate how long it takes your money to grow to a certain amount. You can predict how fast a disease spreads through a population. You can figure out how long a radioactive substance takes to decay. You can estimate how long it takes a cup of hot coffee to cool down to room temperature. All of these come down to understanding that e is the base of continuous growth, and you can work backward using natural logarithm to find the time or the amount or whatever you’re looking for.

The word natural in natural logarithm exists because e is the natural choice for describing growth. It’s not natural because it’s common or easy. It’s natural because anything that grows continuously, without jumping around, will produce e. Nature doesn’t choose e because someone told it to. Nature produces e because e is what the math requires. Every bacteria that divides, every dollar that earns interest, every cup of coffee that cools, every disease that spreads is computing e without knowing it. The math is already there in the physics. We’re just describing what’s already happening.

And the weirdest part is that none of this was planned. John Napier invented logarithms for calculations. Nobody imagined that centuries later his work would lead to a number that shows up in every continuous growth process in the universe. Leonhard Euler was just studying mathematical functions and noticed something appeared repeatedly. He named it e. Then people realized e was the key to understanding everything from population dynamics to finance to physics. A number that wasn’t invented for any purpose became essential to understanding how the world actually works.

Sources and Further Reading

• Demystifying the Natural Logarithm (ln) — BetterExplained
• What’s so special about Euler’s number e? | 3Blue1Brown
• How did the notation “ln” for “log base e” become so pervasive? — Mathematics Stack Exchange
• Where Do Logarithms Come From?