I’m going to explain this number using basic arithmetic, so bear with me.
It’s such an unpopular number that there isn’t even a dedicated place to talk about it — people just get redirected to threads about Euler’s constant instead. That alone says something. What even is math? Addition, subtraction, multiplication, division, geometry, word problems — basic stuff anyone could follow if you sat them down and explained it simply, kid included. But at some point, things get complicated. It’s as if mathematicians deliberately buried the subject under symbols first (summation signs, weird Greek letters), then derivatives, integrals, logarithms, and complex numbers — making sure nobody else could follow along so they could keep the field to themselves. And then, ironically, a lot of them forgot the reasoning behind the formulas too. If they didn’t literally memorize the formula, they’d get it wrong. People internalize formulas through repetition and start believing that means they understand the math.
Even the proof of the formula is so packed with dense notation that you can never fully absorb it. Everyone has a point in math where they lose the thread — mine was e.
Anyway, to keep this simple: e is roughly 2.71828…, an infinite non-repeating decimal (per Wikipedia). Pi is also infinite, but ask anyone and they can immediately tell you what it represents — the ratio of a circle’s circumference to its diameter. Done, simple. So what is e? The most tangible explanation is compound interest.
Say you have 1 lira, and you put it in the bank at 100% annual interest. After a year, with interest, you have 2 lira. Now, if you cut the interest rate but increase how often it compounds at the same overall rate — say, a 6-month rate of 50%, compounded twice a year — after a year you’d have 2.25 lira. If you kept shortening the compounding period toward zero (down to months, days, hours, seconds) while proportionally lowering the rate each time, you’d keep earning slightly more. But there’s a ceiling to this. If you push the compounding frequency to its limit, the maximum amount you can reach converges to that number: 2.7182… It’s a genuinely tangible, real-world example, and yet it still isn’t widely understood.
Here’s what e actually represents: the rate of continuous growth. A bank gives you a nice, clean monthly or annual interest rate. But in nature, plenty of things grow continuously — not in discrete jumps. Say a certain type of bacteria doubles every 5 minutes. If you start with 1 million of these cells and say the growth rate is “100% every 5 minutes,” you can’t calculate this the way you’d calculate bank interest, because the growth is happening continuously, not step by step — the number of cells able to reproduce keeps increasing every fraction of a second. The moment you write down “I have 1 million cells,” there are already more in the dish. So for anything with an extremely short “compounding period,” you use e.
If the growth rate is 100% per 5 minutes, then after roughly 5 minutes you’d have about 2.71 million cells (not exactly 2 million, because the growth happens continuously). After another 5 minutes, it’s e × e. After 15 minutes total, e × e × e. And so on. What if it’s 17 minutes instead of 15? What if the growth rate is 50% instead of 100%? Mathematicians, never ones to leave a formula alone, turned all of this into something needlessly obscure — dragging natural logarithms into it — when really: actual growth = e^(growth rate × number of compounding periods). Instead of just saying that plainly, they buried it in logarithmic notation. And when you’re cramming for exams, you never actually get to dig into the “why” — I only really understood this myself very recently.
In short: e is the number used to measure the rate of growth for anything where the number of compounding periods (the frequency of discrete growth steps) gets extremely large.
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