e Is Where You Land If You Don’t Turn

Students meet e twice, and both times it is defined by its behavior. At the bank: compound interest more and more often and (1 + 1/n)ⁿ creeps toward 2.718… In calculus: e^x is the function that is its own derivative. Both are true. Both are also strangely indirect — they describe what e does, never what e is. A student can pass both courses and still hold no picture of the number itself, the way they hold a picture of 2, or of π.

In the last lesson we built a circle from one rule: each new arrow is the last one times iθ/n — turn 90°, shrink, repeat. The circle was not an ingredient; it was the destination. This lesson asks the obvious next question: what happens if you keep the rule and delete the turn?

Rotation and growth are the same machine. The only difference is whether the steps turn.

Keep everything from lesson one except the i. Each arrow is now the previous arrow times a plain number:

No turning means every arrow points the same way — due east along the number line. Start with an arrow of length 1, apply the rule, and lay the arrows tip to tail. At x = 1 the steps are 1, 1, ½, 1/6, 1/24, 1/120… and the running total marches 1, 2, 2.5, 2.667, 2.708, 2.717, 2.7181… The walk is heading somewhere specific. It lands at 2.71828… — and that landing point is e. Not a rate, not a limit-of-a-process at the bank: a position on the number line, reached by the same engine that drew the circle, pointed straight.

Written as a formula, the walk is exactly the exponential series with the i stripped out:

And the rule still works pointed backwards. Set x = −1 and the arrows alternate east, west, east, west: the total bounces 1, 0, 0.5, 0.333, 0.375, 0.3667… and settles on 0.36788 = 1/e, each bounce bracketing the answer from the other side. One engine, three destinations so far: point the steps with a turn and you get the circle; point them forward and you get e; alternate them and you get 1/e.

The left pane is the walk: pick x, add arrows one at a time, and watch the total settle. Try x = 1 first, then x = −1 for the bounce, then x = 2 to feel how a longer walk needs more steps. Press reveal e^x only after the class has guessed where the walk is heading. The right pane answers last lesson’s open question — why fourteen terms? — and is where e shows up a second time, in a place no bank ever mentions.

Here is what the compound-interest story was hiding. Expand (1 + 1/n)ⁿ with the binomial theorem and you get 1 + 1 + a slightly shrunken ½ + a more shrunken 1/6… — the same arrows, each multiplied by a factor just under one that melts away as n grows. Continuous compounding was never a different idea from the series. It is the same walk, taken through a fog that thins as n → ∞. The bank creeps; the walk pounces:

The second story they know — e^x is its own derivative — is usually delivered as an algebraic fact to memorize. The walk makes it visible. Differentiate the arrows one at a time: the derivative of xⁿ/n! is xⁿ⁻¹/(n−1)!, which is exactly the previous arrow. Differentiation does not mangle the walk; it shifts it one step left. The constant 1 falls off the front, every other arrow slides into its neighbor’s place, and the infinite walk comes out identical to itself.

This is why the walk is self-propelled: at every moment, its rate of growth is its current size, because its list of parts is its own list of rates. A student who sees the shift once will never again need to memorize which function survives differentiation.

Now the debt from lesson one. The circle needed about fourteen terms, and the reason had a number hiding in it: convergence turns decisively fast once n passes about 2.7×θ. That constant was not approximate folklore. It was e, moonlighting.

Look at the lengths of the arrows a walk of size θ uses: θⁿ/n!. Each is the previous one times θ/n — so the lengths grow while n < θ, peak at n ≈ θ, and then shrink. But shrinking is not enough; the question is when the shrinking becomes irreversible collapse. The answer, which falls out of Stirling’s approximation n! ≈ (n/e)ⁿ√(2πn), is n ≈ e·θ: past that point each new arrow is smaller than the last by a factor of e or better, and every further step multiplies the remaining error by less and less. e is the exchange rate between distance and steps — ask the walk to go θ far, and expect to pay about e·θ steps before the answer locks in.

There is one more place the same fact hides, almost too tidy to be true. Take all the whole numbers from 1 to N and ask for their typical size — the geometric mean of 1×2×⋯×N. For large N the answer approaches N/e. For N = 1000 the true value is 369.5 against N/e = 367.9. The factorial — the machine that beat the powers, closed the circle, and paid for every walk on this page — carries e inside it as its own average. The number the bank creeps toward and the number the factorial is made of are the same number, and neither story tells you that. The walk does.