Imaginary interest rates | Ep. 5 Lockdown live math

An “imaginary interest rate” isn’t just a math prank: when interest compounds continuously, an interest rate of √−1 turns money growth into circular motion in the complex plane—so the real part oscillates between positive and negative values instead of steadily increasing. The punchline is practical: if a bank offered an annual rate of √−1 and compounded continuously, the account would repeatedly return to its starting value (after multiples of 2π years), but it would also swing negative in between, making it unusable for real life.

The lesson starts by building intuition for ordinary compound interest and why compounding frequency matters. A bank paying 12% per year makes a balance jump from $100 to $112 only at year’s end, but the key mechanism is multiplicative growth: each time interest is applied, the next interest payment is proportional to the new, larger balance. That’s why 12% for 10 years yields about $310, not $220. When compounding happens more frequently—like 6% every six months or 1% each month—the balance ends up higher, because the “rate of change is proportional to what you already have” keeps feeding back into itself. The Desmos walkthrough frames this with repeated multiplication by constants like 1.12 or 1.01, and then generalizes to a limit form where compounding becomes continuous.

That generalization leads to the constant e. By slicing time into n tiny intervals, the balance after time t becomes proportional to (1 + r·t/n)^n. As n grows, this expression approaches a constant: e ≈ 2.71828. In real-valued interest, this produces continuously compounded growth of the form e^(rt), tying the limit definition of e directly to the familiar exponential law.

The “imaginary rate” question then reuses the same limit machinery, but with r = √−1. Plugging √−1 into e^(rt) doesn’t make sense if interpreted as ordinary real exponentiation, yet the original limit definition still works because it only relies on algebraic operations that complex numbers support. Geometrically, multiplying by i rotates a vector by 90 degrees. So each tiny compounding step adds an increment perpendicular to the current “money vector,” meaning the balance doesn’t grow in magnitude the way real interest does—it moves around a circle.

To show how compounding frequency changes outcomes, the lesson contrasts annual compounding (big 90-degree jumps) with continuous compounding (infinitesimal steps). With annual steps, the real part becomes negative after enough years, and the account can owe money at certain times. Under continuous compounding, the motion becomes a smooth rotation: after π/2 years the balance is purely imaginary; after π years it’s purely negative real; after 2π years it returns to the starting value. The bank’s offer is therefore a “hold-and-wait” scenario only in the abstract sense—emotionally and financially it’s unstable.

Finally, the same mathematics becomes physics. By packaging spring displacement x and velocity v into a complex number, Hooke’s law (force = −kx) turns the dynamics into a 90-degree-rotation rule. The result is simple harmonic motion: displacement follows cosine behavior and velocity follows a phase-shifted sine behavior, all interpretable as shadows of circular motion. The imaginary-interest-rate thought experiment thus lands on a real-world endpoint: the exponential-of-imaginary form (via Euler’s formula) is exactly what describes oscillations without friction.