Euler's Number e: The Constant of Continuous Growth
Imagine you find a bank offering 100% annual interest. Good deal. Now they offer to compound it monthly — so each month you earn 1/12 of 100%, applied to your growing balance. Better. Then weekly. Then daily. Then every second. Then every nanosecond. At some point you'd expect the interest to grow without bound. But it doesn't. No matter how frantically you compound, the growth converges to a fixed ceiling: approximately $2.718 for every $1 invested. That ceiling is e — Euler's number — and the fact that it exists at all is one of the more quietly astonishing facts in mathematics.
The Concept
e is a number, approximately 2.71828182845904..., that arises naturally whenever something grows (or decays) at a rate proportional to itself. Population doubling. Radioactive atoms disintegrating. A capacitor charging through a resistor. Coffee cooling on your desk. All of these processes are secretly governed by e.
There are several equivalent ways to define it. Jacob Bernoulli's original formulation from 1683 is the most intuitive:
e = lim (1 + 1/n)^n as n approaches infinity
Plug in a few values and watch it converge:
- n = 1: (1 + 1)^1 = 2
- n = 10: (1.1)^10 ≈ 2.5937
- n = 100: (1.01)^100 ≈ 2.7048
- n = 1,000: (1.001)^1000 ≈ 2.7169
- n = 1,000,000: ≈ 2.71828
It converges — slowly, but surely — to e.
The second definition comes from an infinite series. Add up the reciprocals of every factorial:
e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + ... e = 1 + 1 + 0.5 + 0.1667 + 0.0417 + 0.0083 + ...
Each term shrinks rapidly (factorials blow up fast), and the sum lands exactly on e. This series converges so quickly it's one of the easiest ways to actually compute e to thousands of decimal places.
The third definition — the one that makes calculus work — is perhaps the deepest: e is the unique positive number whose exponential function is its own derivative.
For the function f(x) = e^x, the slope at every point equals the value at that point. If you're at height 7.389 on the curve (which is e²), the slope there is also 7.389. This self-referential property is what makes e the natural base for describing any process where change depends on current state.
Why It Matters
The mathematics of growth and decay
The equation A(t) = A₀ · e^(rt) describes exponential growth, where A₀ is the starting amount, r is the growth rate, and t is time. Plug in a negative r and you get exponential decay. This single formula covers:
Population biology: A colony of bacteria doubling every 20 minutes follows e^(rt) precisely. Ecologists use it to model everything from algae blooms to pandemic spread.
Radioactive decay: Carbon-14 decays as N(t) = N₀ · e^(-λt). This is how archaeologists date ancient wood, bone, and charcoal — they measure how much C-14 remains and solve for t. Carbon dating is grounded entirely in e.
Finance: Continuous compounding uses A = Pe^(rt). If you invest $1,000 at 5% continuously compounded for 10 years, you get $1,000 · e^(0.5) ≈ $1,648.72. Banks and investment firms use this formula for real loan calculations.
Cooling: Newton's Law of Cooling says the temperature difference between an object and its surroundings decays as e^(-kt). That cup of coffee really does cool along an e-curve.
Why e is the natural base for calculus
In high school you may have learned logarithms in base 10. But mathematicians and physicists almost always work in base e — the natural logarithm, written ln(x). Why?
Consider a general exponential function y = a^x. Its derivative turns out to be y' = a^x · ln(a). There's always this extra ln(a) factor — except when a = e, because ln(e) = 1. Base e is the only base where the derivative of a^x is exactly a^x, no extra factor needed. Every other base introduces a constant multiplier that clutters every equation. Nature's math is written in base e.
Euler's identity: the most beautiful equation
The number e has a surprising relationship with circles. Euler's formula states:
e^(ix) = cos(x) + i·sin(x)
where i = √(-1). Setting x = π gives the famous Euler's identity:
e^(iπ) + 1 = 0
Five fundamental constants — e, i, π, 1, and 0 — connected in a single equation. Three basic operations — exponentiation, multiplication, addition — each appearing exactly once. Mathematicians have been calling this the most beautiful equation in mathematics for nearly 300 years, and the label has stuck.
The Details
Who discovered it and when
The story begins with John Napier (1550–1617), who invented logarithms in the early 1600s. His tables implicitly pointed toward a special base, but he never isolated the constant itself.
In 1683, the Swiss mathematician Jacob Bernoulli was studying the compound interest problem — the same one we opened with — and found the limit of (1 + 1/n)^n. He didn't name it or give it a symbol; he just noted it was somewhere between 2 and 3.
Gottfried Leibniz wrote to Christiaan Huygens in 1690 using the letter "b" for this constant. But the name and symbol we use today came from Leonhard Euler, the most prolific mathematician in history. Euler first used the letter "e" in an unpublished paper around 1727–1728, then in a letter to Christian Goldbach on November 25, 1731. His published use appeared in Mechanica in 1736. By 1748, in his masterwork Introductio in Analysin Infinitorum, Euler had fully established the series expansion, the limit definition, and the central role of e in analysis.
Why "e"? Nobody knows for certain. Euler was already using "a" in his equations, so "e" was simply the next available vowel. The letter stuck because Euler's influence was so dominant that his notation became everyone's notation.
e is irrational and transcendental
Euler himself proved in 1737 that e is irrational — it cannot be written as a fraction p/q. His proof used the continued fraction representation of e, which is infinite and non-repeating. Rational numbers always have terminating or repeating continued fractions, so e must be irrational.
But e is something even stranger: it's transcendental. In 1873, the French mathematician Charles Hermite proved that e cannot be the solution to any polynomial equation with rational coefficients. Numbers like √2 are irrational but still algebraic (√2 satisfies x² - 2 = 0). Transcendental numbers lie entirely outside the reach of polynomial algebra.
Hermite's proof of e's transcendence was the first time a number that arose naturally from real-world processes — not one specifically constructed to be weird — was proved transcendental. It was a landmark result, and his technique was later extended by Ferdinand von Lindemann in 1882 to prove that π is also transcendental.
Think about what that means: a number first noticed because of bank interest turns out to be fundamentally unreachable by ordinary algebra. The deeper you dig, the stranger e becomes.
The secretary problem: 37% and the art of knowing when to stop
Here's a puzzle. You're interviewing n candidates for a job, one by one, in random order. After each interview you must immediately decide: hire this person, or pass. You can't go back. What's your optimal strategy?
Mathematics says: interview the first 1/e ≈ 36.8% of candidates purely as a benchmark — hire none of them. Then hire the first candidate who is better than everyone you've seen so far. If you follow this rule, your probability of landing the best candidate is approximately 1/e ≈ 36.8%.
Two appearances of 1/e in one problem. This isn't a coincidence — it's a consequence of how e governs probability when choices are irreversible. The strategy applies anywhere you're making sequential decisions with no going back: house hunting, hiring, online dating, even picking a parking spot. Skip the first 37%, then commit to the next best thing you see.
e in electronics: RC circuits
If you've studied basic circuits, you've seen e without knowing it. When a capacitor charges through a resistor, the voltage across the capacitor follows:
V(t) = V₀(1 − e^(−t/RC))
Where R is resistance, C is capacitance, and RC is the "time constant" τ (tau). After one time constant, the capacitor has charged to 1 − e^(−1) ≈ 63.2% of its final voltage. After five time constants, it's at 99.3%. Engineers use this 63.2% benchmark constantly when designing filters, power supplies, and timing circuits. It's e, showing up in your phone's charging circuit.
e in probability: the Poisson distribution
Suppose customers arrive at a coffee shop at an average rate of 3 per hour. What's the probability that exactly 5 arrive in the next hour? The Poisson distribution gives the answer:
P(k events) = (e^(−λ) · λ^k) / k!
With λ = 3 and k = 5: P = (e^(−3) · 3^5) / 5! ≈ 10.1%.
The e^(−λ) term represents the probability of zero events occurring — the baseline from which all other probabilities build. This distribution models phone calls arriving at a switchboard, mutations in a strand of DNA, cars passing through an intersection, shooting stars appearing in a given hour. Wherever rare events happen at a known average rate, e is hiding in the math.
How many digits do we know?
As of 2020, e has been computed to over 31 trillion decimal places — specifically 31,415,926,535,897 digits, a figure that is itself a playful nod to π. In practice, a few dozen digits suffice for any engineering or scientific purpose. But computing trillions of digits tests both the limits of computer hardware and the efficiency of numerical algorithms. It's less about the digits themselves and more about the sport of reaching them.
On the human side, the memorization record as of 2024 stands at 14,000 digits, set by Deepu V of India.
Takeaways
- e ≈ 2.71828 emerges from the compound interest limit (1 + 1/n)^n as n → ∞, and is equivalently defined by the factorial series 1 + 1/1! + 1/2! + 1/3! + ...
- e is the natural base for growth and decay because it's the only base where the derivative of a^x equals a^x itself — no extra constant factor, no clutter. Every process where rate-of-change is proportional to current size is naturally described by e.
- e is irrational and transcendental, proved by Euler in 1737 and Hermite in 1873 respectively — meaning it can't be expressed as a fraction or as the root of any polynomial with rational coefficients.
- e appears everywhere unexpected: in the optimal job-interview strategy (skip the first 37%), in how capacitors charge in electronics, in the Poisson distribution for rare-event probability, and in Euler's identity connecting e, i, π, 0, and 1 in a single equation.
- The history spans 350+ years: from Bernoulli's 1683 compound interest curiosity, through Euler's formalization in the 1730s and 1740s, to Hermite's 1873 transcendence proof — each era discovering that e was even more fundamental than the last.
If there is one number that sits at the intersection of finance, physics, probability, and pure mathematics, it's e. The fact that it first appeared in a problem about bank interest — of all things — and turned out to be transcendental, irreducible, and woven into the fabric of calculus itself, is exactly the kind of surprise that makes mathematics worth paying attention to.