The number e is about 2.71828 because it is the limit of (1 + 1/n)n: what one dollar becomes in a year at 100% interest when the interest is paid more and more often. Nobody chose that number; the arithmetic forces it. The same value is the only base whose exponential curve has a slope equal to its own height everywhere, and the point where the area under 1/x, starting at 1, reaches exactly 1. Its digits come from the sum 1 + 1 + ½ + ⅙ + 1/24 + ⋯, and nothing rounds them.
We discovered it, like a mountain, and it started with a greedy question about money.
A bank offers you a deal: 100% interest, one year, on $1. If the interest is paid once, at year’s end, you finish with $2. Now split the deal: 50% after six months, then 50% on whatever you hold. At six months you have $1.50, and those extra 50 cents now earn interest too. 1.5 × 1.5 gives $2.25. Twice beats once.
Keep splitting, and the balance keeps rising:
| Interest paid | Balance after one year |
|---|---|
| once | $2 |
| twice | $2.25 |
| quarterly, 25% each time (1.25⁴) | about $2.4414 |
| monthly | $2.6130 |
| daily | about $2.71457 |
| hourly | $2.71813 |
| every second | about $2.718282 |
The balance approaches something, not infinity. Yearly to half-yearly gains 25 cents; half-yearly to quarterly, about 19; monthly to daily, about 10. The sequence rises but never passes 3, so it must converge. Underneath is a tug of war: each factor, 1 + 1/n, shrinks toward 1, but the number of factors grows. Neither side wins outright. The paired Short runs the same compounding in under a minute: Compound interest and the number e.
Bernoulli’s question, Euler’s letter
In 1683, Jacob Bernoulli studied this compound-interest problem, looking for the limit of (1 + 1/n)n as n grows. Using the binomial theorem, he showed that the limit had to lie between 2 and 3, without finding its full value. His discussion of interest appeared in the journal Acta Eruditorum in 1690, the same year Gottfried Leibniz, writing to Christiaan Huygens, used the letter b for the number.
Leonhard Euler supplied its lasting name: e, used in a letter to Christian Goldbach dated 25 November 1731. He had already written e this way in an unpublished paper on the explosive force of cannon fire, from 1727 or 1728, and the letter first appeared in print in his Mechanica of 1736.
In 1737, Euler proved that e is irrational: no fraction of whole numbers equals it, and its decimal never settles into repetition. (The proof was published seven years later.) In 1873, Charles Hermite proved more: e is transcendental. No nonzero polynomial with integer coefficients has e as a root, so nothing like x² = 2 can capture it. It was the first number proved transcendental without being built for the purpose. Bernoulli posed the question, Euler named the answer, and Hermite established its algebraic boundary.
Why e is 2.718…, digit by digit
Expand (1 + 1/n)n with the binomial theorem and watch what the multiplication leaves behind: 1, plus 1, plus something just under ½, plus something just under ⅙, and onward. The third term, for example, is n(n − 1)/2 times 1/n², which is ½ × (1 − 1/n): a little less than a half. As n grows without bound, each term approaches the reciprocal of a factorial, and the limiting sum is
1 + 1 + ½ + ⅙ + 1/24 + ⋯ = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ⋯
Factorials explode, so their reciprocals shrink fast. Ten terms already give about seven significant digits. Compare the tail with a halving series, ½ + ¼ + ⅛ + ⋯: term by term it is no bigger, and the halving series totals 1, so the tail totals less than 1. That keeps the whole sum below 3. After the two 1s, the actual remainder is about 0.718.
That is the answer to why e is 2.718 and not something rounder. The number is not round because nothing rounded it. It is exactly what the arithmetic leaves behind.
The slope that equals itself
Forget money. Draw y = 2x and, at every point, measure the curve’s steepness. Plot those slopes and another exponential appears, with every height scaled down by about 0.693. Now try 3x: its slope curve stands taller, scaled up by about 1.0986. One base undershoots itself; the other overshoots. (Those two factors are the natural logarithms of 2 and 3.)
Between 2 and 3, exactly one base makes the slope curve land on the original, with scale factor 1. Narrow it down and out comes 2.71828…, the bank’s number again. That is not a coincidence; it is the same limit in a different disguise. It gives a second definition: ex is the function whose rate of change equals itself, starting at 1. At 0 it has height 1 and slope 1, and everywhere else its slope equals its height.
Area gives a third. Draw 1/x and shade beneath it, starting at 1. How far right must you go before the shaded area equals 1? The endpoint is e. These are three views of one structure, not three unrelated tricks. Continuous interest grows in proportion to the money already there, and at 100% a year its growth rate equals the balance itself. e is not a fact about banking. It is a fact about change.
Where e turns up
Flip the sign and growth becomes decay. A fixed proportional loss gives the factor e−kt, which models radioactive populations, capacitor discharge, and a cooling coffee’s temperature difference from its surroundings. The ideal curve approaches zero without reaching it.
Chance works the same way. Play an independent lottery a million times, with a one-in-a-million chance each time. The chance of never winning is (1 − 1/n)n with n = 1,000,000, and as n grows that approaches 1/e, about 37%: the bank’s limit, mirrored. Hand back coats at random, and the chance that nobody gets their own is also close to 1/e; with 5 coats or 5,000,000, the answers barely differ. That puzzle has its own story: Four letters, four envelopes.
The bell curve of random error is shaped by e−x²/2. And in the classic selection problem, where randomly ordered candidates arrive once with no recalls, the rule for many candidates is to inspect the first 37%, then choose the next one who beats everyone seen. Different settings, recurring structures: proportional change, repeated tiny chances, factorials.
The circular walk
One last question: what could e to an imaginary power mean? Extend the same rule. For eit, the rate of change is i times the current value. Multiplying by i makes a quarter turn, so the velocity always points at right angles to the position. Motion perpendicular to the position preserves distance from the origin, so a point starting at 1 stays on the unit circle. Its speed is also 1, so in time t it covers t radians. The video draws it as a point sweeping out a semicircle.
Walk π radians, halfway around, and you land on −1:
It is not mysticism. It is compound interest’s number taking a circular walk, and the walk is a half turn, which is exactly what π measures: see π is the unit of turning.
e was never chosen. Let growth feed on itself, and 2.718281828… answers: in banks, in atoms, in luck, and in a circle.
