Euler's Number- Khan Academy Explains e
What the Heck Is Euler's Number?
You're probably familiar with π (pi) — that 3.14159 number that keeps showing up in circles. Well, e is another constant like that, approximately 2.71828...
But here's what trips people up: unlike pi, which has a geometric meaning anyone can visualize, e comes from a calculus concept. That makes it harder to "get" intuitively. Khan Academy tries to fix this, and I'll break down their approach.
Where e Actually Comes From
The formal definition is dry as dust:
e is the limit of (1 + 1/n)^n as n approaches infinity
Yeah, that doesn't help much. Let me give you something useful instead.
The Compound Interest Version
Imagine you put $1 in a bank account with 100% annual interest. Basic math says you'll have $2 after a year.
But what if interest compounds more frequently? Here's what happens:
- Compounded annually: $1 × (1 + 1) = $2.00
- Compounded semiannually: $1 × (1 + 0.5)^2 = $2.25
- Compounded monthly: $1 × (1 + 1/12)^12 = $2.61
- Compounded daily: $1 × (1 + 1/365)^365 = $2.71
- Compounded continuously: $1 × e^1 = $2.718...
That "compounded continuously" result? That's e. It represents the maximum possible growth from continuous compounding — you can't beat it no matter how finely you slice the time periods.
How Khan Academy Explains It
Khan Academy's approach starts with the (1 + 1/n)^n formula and walks you through what happens as n gets huge. Sal Khan (the founder) has a habit of building intuition through examples before touching the formal math.
His calculus playlist covers:
- Introduction to e through continuous compound interest
- Understanding the derivative of e^x
- Why e^x is its own derivative (this is the big one)
- Natural logarithm properties tied to e
The good news: if you've watched his videos on exponential functions, you're halfway there. e is just a specific exponential base — one that shows up naturally in calculus because of its derivative property.
The Derivative Property That Makes e Special
Most exponential functions are annoying in calculus. The derivative of 2^x involves a messy limit. But e^x just... differentiates to itself:
d/dx(e^x) = e^x
This makes the math cleaner. That's not a coincidence — it's why e is called the natural base. Calculus problems involving growth and decay naturally land on e because it simplifies everything.
e Shows Up Everywhere
Once you know to look for it, you'll see e in places you didn't expect:
- Normal distribution — that bell curve in statistics? Its formula uses e
- Radioactive decay — physics problems involving half-lives use e
- Population growth models — biology uses e constantly
- Probability — the Poisson distribution, waiting time problems
- Signal processing — complex exponentials with e^x describe waves
It's not magic. It's just that continuous processes — which show up constantly in science and engineering — naturally involve e.
Comparing: e vs π vs Other Constants
| Constant | Approximate Value | Where It Comes From | Primary Use |
|---|---|---|---|
| e | 2.71828... | Calculus / limits | Continuous growth, decay, probability |
| π | 3.14159... | Geometry / circles | Circles, angles, periodic phenomena |
| φ (golden ratio) | 1.61803... | Algebra / geometry | Fibonacci sequences, art, architecture |
| √2 | 1.41421... | Geometry / diagonals | Right triangles, diagonal calculations |
e is the only one of these that calculus produces on its own. The others come from geometry or algebra. That's why mathematicians call it the natural base — it emerges from the mathematics of change itself.
Getting Started: How to Actually Learn This
Skip the Wikipedia rabbit hole. Here's a practical path:
- Watch Khan Academy's introduction to e — search "Euler's number introduction" in their calculus playlist. Sal walks through the compound interest example, which is the most intuitive entry point.
- Understand the limit visually — use a spreadsheet. Plug in larger and larger values of n into (1 + 1/n)^n. Watch it converge toward 2.71828.
- Learn e^x derivatives — this is where Khan Academy's calculus content shines. The fact that e^x is its own derivative is the key property you need for exams.
- Connect to ln(e) — the natural logarithm ln(x) is the inverse of e^x. If you understand logarithms, this clicks fast.
The Bottom Line
e isn't complicated because it's hard — it's complicated because most textbooks introduce it backwards. They start with the formula and expect you to feel its significance.
Khan Academy does it right: start with compound interest, show why the number matters, then introduce the formal definition. Once you see e as "the growth constant that differentiates cleanly," everything clicks.
Go watch the videos. Then come back and look at the formula (1 + 1/n)^n — it'll make sense this time.