What Is Euler’s Number (e) in Math? Why It Matters for Growth, Calculus & Exams
Picture this: you're sitting at your desk staring at your graphing calculator before a big AP Calculus exam, and right there on the keypad is the letter e. Not x. Not y. The letter e. And unlike every other letter in math class, this one isn't a variable you solve for — it's something else entirely. Something specific. Something that has puzzled students for generations, even as it quietly powers some of the most elegant mathematics in existence.
So, what is Euler's number? It's a question we field constantly at HYE Tutors, and for good reason. Euler's number e appears across calculus, compound interest, population growth, radioactive decay, and even the equations behind the COVID-19 spread models you might have seen in the news. In other words: it's everywhere.
As a team of tutors with affiliations at Columbia, UC Berkeley, and UCLA — with over five years of one-on-one experience guiding students through AP Calculus, SAT prep, and college-level math — we've walked hundreds of students through this very moment of confusion. The good news? Once e clicks, it really clicks. And by the end of this guide, you'll understand what e is, where it comes from, why it matters for your exams, and how to use it confidently.
“Euler’s number, written as e, is an irrational mathematical constant approximately equal to 2.71828. It is the base of the natural logarithm and arises naturally when calculating continuous growth or decay. Like π (pi), e never ends and never repeats. Named after Swiss mathematician Leonhard Euler, it is one of the most important numbers in mathematics — appearing in calculus, finance, biology, and physics.”
What Is Euler's Number (e) in Math?
In our five years of tutoring, Euler's number has appeared on AP Calculus AB exams, AP Calculus BC free-response sections, ACT Math, and even in high school biology labs. We've worked with students who glossed over e for an entire semester — treating it vaguely as "some number" — and then watched them struggle on problems that should have been straightforward. We've also worked with students who took the time to genuinely understand e, and those students almost always perform better across the board. So let's build that understanding from the ground up.
Euler's Number Definition — What Does e Mean in Math?
Let's start simple. Euler's number, written as e, is a mathematical constant — a fixed, specific value that never changes, just like π (pi). It is not a variable. You cannot solve for it. It simply is, and its approximate value is:
e ≈ 2.71828182845…
The decimal goes on forever without repeating — which is what makes e an irrational number. It is also transcendental, meaning it is not the root of any polynomial with rational coefficients (a detail mostly relevant at the college level, so don't worry too much about that for now).
Euler's number is the base of the natural logarithm, written as ln. Just as π describes the geometry of circles, e describes the mathematics of continuous growth and change. And one quick pronunciation note before we go further: it's named after Leonhard Euler, whose name is pronounced "Oiler" — not "Yooler." We've made this correction in tutoring sessions more times than we can count, so consider yourself ahead of the curve.
The practical takeaway for most students: you don't need to memorize all the decimals. For AP Calculus and ACT Math, knowing that e ≈ 2.718 and being able to use the eˣ key on your calculator will take you very far.
Where Does Euler's Number Come From? (The Origin of e)
This is our favorite part to teach, because the origin of e isn't some abstract mathematical accident — it comes from a beautifully concrete question about money.
Imagine you deposit $1 in a bank account that pays 100% annual interest. Simple enough. If the bank compounds interest once per year, you end the year with $2.00. Not bad. But what if the bank compounds more frequently?
Compound once (annually): $2.00
Compound twice (every 6 months): $2.25
Compound monthly (12 times): ≈ $2.613
Compound daily (365 times): ≈ $2.7145
Compound continuously (infinite times): ≈ $2.71828… → this is e
As the compounding becomes more and more frequent — approaching infinity — the result converges on a single, specific number. That number is e. Mathematically, this is expressed as:
e = lim(n→∞) (1 + 1/n)ⁿ
The key insight: e is the mathematical result of infinite compounding growth. It's not arbitrary — it's the number that nature itself keeps arriving at when growth happens continuously. That's why mathematicians call it nature's growth constant.
We always use the bank account story in our first session on e. Without fail, every student gets it immediately — and once you understand the origin of e, the formula stops feeling like something to memorize and starts feeling inevitable.
What Is the Value of Euler's Number?
The value of e is approximately 2.71828 — that's the figure you'll see on most exams and in most textbooks. Here's what students need to know practically:
For AP Calculus and ACT Math, knowing e ≈ 2.718 is almost always sufficient
On your graphing calculator (TI-84), press 2nd then LN to access the eˣ function
e is irrational: its decimal expansion is infinite and non-repeating
e is transcendental: it cannot be expressed as the root of any rational polynomial (college-level detail)
Euler's Number vs. Euler's Formula — Are They Different?
Yes, and this distinction trips up a lot of curious students. Euler's number is the constant e ≈ 2.71828, which is what this entire article is about. Euler's formula is a separate (though related) equation from complex analysis:
e^(iπ) + 1 = 0
This equation — often called Euler's identity — is frequently described as "the most beautiful equation in mathematics" because it connects five of the most fundamental constants: e, π, i (the imaginary unit), 1, and 0. It's genuinely stunning.
But here's the practical note: Euler's identity is a college-level or advanced pre-calculus topic. You won't need it for AP Calculus AB, SAT, or ACT. We mention it because at least one student per semester asks us about the "beautiful equation" they saw online — now you have context without the confusion.
Properties of Euler's Number — What Makes e Special?
So why do mathematicians and scientists keep choosing e over other numbers? Because e has a property that is genuinely unique in all of mathematics:
The derivative of eˣ is eˣ itself.
In plain English: if you measure the rate of change of the function eˣ at any point, you get eˣ again. The function is its own derivative. No other base has this property — and it's not a coincidence. It's the reason calculus textbooks reach for eˣ constantly. A few other notable properties:
The natural logarithm (ln) uses e as its base: ln(e) = 1
e appears inside the normal distribution (bell curve) formula used across statistics and science
It appears in Euler's identity, complex analysis, and differential equations at the college level
Students going on to study physics, engineering, biology, economics, or psychology will encounter e in virtually every quantitative course
The derivative property alone is why calculus textbooks love eˣ. It behaves in a way no other base does — and understanding that is a genuine intellectual advantage for students.
The Natural Exponential Function — What Is eˣ?
When you see eˣ in a math problem, that notation means "e raised to the power of x." It is a function — not just a single number — and it's the most commonly used exponential function in calculus and science.
Here's how it behaves:
eˣ is always positive, no matter what value x takes
It is always increasing — it never levels off or turns downward
Its slope at any point equals its value at that point (the derivative property in action)
Graphically, it starts near zero for large negative x values and grows rapidly as x increases
Compare this to other exponential functions like 2ˣ or 10ˣ: they grow similarly, but when you try to differentiate them, you get extra awkward factors (like ln(2) or ln(10)) attached to the result. With eˣ, the derivative is clean, elegant, and — crucially — identical to the original function. That's why calculus prefers e above all other bases.
If you're interested in other famous mathematical constants and the surprising places they appear, our guide on the Golden Ratio in Math explores another remarkable number with deep roots in nature and geometry.
