Imaginary Interest Rates, Complex Numbers and Springs

What would it mean for a bank to offer an interest rate of i, the square root of −1? Grant Sanderson uses this deliberately strange question to connect compound interest, complex numbers and harmonic motion.

First consider ordinary interest. Bank A increases $100 by 12% once a year; bank B increases it by 1% each month. Although twelve monthly rates add to 12%, compounding means the two outcomes differ.

With annual compounding the balance becomes $112 after one year. The increase can be expressed as multiplication by 1.12; after a second year, interest is earned on the first year’s interest too.

For two six-month periods at 6% each, the balance is 100 × 1.06². Each step multiplies the current balance by the same factor, so repeated percentage increases naturally become powers.

Monthly compounding gives 100 × 1.01¹², approximately $112.68: about 68 cents more than annual compounding. The difference becomes more visible over longer periods.

Let the initial amount be M₀, the annual rate r, total time t and the number of steps n. A step of length Δt = t/n changes the balance by rΔt times the current balance.

Factoring gives M → M(1 + rΔt), and repeating n steps gives M₀(1 + rt/n)ⁿ. This expression lets us ask what happens when compounding intervals get finer and finer.

More frequent compounding raises the final balance, but the increase does not grow without bound. For $100 at a 12% nominal annual rate, the continuous limit is about $112.75. Numerical experiments suggest this limit but do not prove it.

An audience question asks whether quaternions arose from the need to solve a polynomial, as complex numbers can arise from x² = −1. Sanderson says their history is instead tied to describing three-dimensional motion and rotations.

The standard limit (1 + 1/n)ⁿ approaches e, approximately 2.71828. With x = rt, the corresponding continuous growth expression is M₀eʳᵗ; the interest-rate problem supplies an intuition for this constant.

The exponential function can also be written as the infinite series 1 + x + x²/2! + x³/3! + …. Sanderson suggests expanding the finite compounding expression as an exercise to see why the two descriptions connect, while noting that a rigorous proof needs care.

Now set the hypothetical rate to i. Reading eⁱᵗ as multiplying e by itself an imaginary number of times is unhelpful, but (1 + it/n)ⁿ still uses familiar operations on complex numbers.

On the complex plane, multiplication by i rotates a vector by 90 degrees. Each small interest increment points perpendicular to the current balance; adding it moves the balance to a new complex value.

The compounding schedule matters. If an initial $1 is compounded once a year at rate i, it becomes 1 + i after one year and 2i after two. After four years it is −4, while after eight years it is 16.

As the steps become infinitesimal, the repeated perpendicular changes approach circular motion rather than an outward spiral. The continuous expression eⁱᵗ traces the unit circle, with t acting as the angle in radians.

Starting with a real dollar, after π/2 units of time the value is i; after π it is −1; after 2π it returns to 1. The banking story is fanciful, but the circular behavior is a useful mathematical model.

The same structure appears in an ideal spring. Hooke’s law approximates the restoring force as F = −kx: the farther the mass moves from equilibrium, the stronger the force pulling it back, within the model’s range of validity.

Newton’s law F = ma turns this into acceleration a = −(k/m)x. The position influences acceleration, acceleration changes velocity, and velocity changes position.

A side question concerns rotations and quaternions. A rotation in two dimensions has one angular degree of freedom; in three dimensions, an axis needs two parameters and the angle a third. Unit quaternions provide a convenient higher-dimensional representation.

The lecture also says four-dimensional rotations have ten degrees of freedom. The correct count for rotations alone is six: an n-dimensional rotation has n(n−1)/2 independent parameters.

Choosing units so that k/m = 1 simplifies the equations to dx/dt = v and dv/dt = −x. Over a small interval, Δx ≈ vΔt and Δv ≈ −xΔt.

Represent the state as a point (x, v). Its small change is proportional to (v, −x), perpendicular to the state vector. Thus position and velocity together trace a circle in this normalized phase space.

Package the pair into z = x + iv. Then dz/dt = −iz, because −i(x + iv) = v − ix. The spring’s coupled equations have become the same kind of complex growth equation as imaginary compound interest, with the opposite direction of rotation.

The solution is z(t) = z(0)e⁻ⁱᵗ. In the ideal frictionless model, the state continually circles; real and imaginary components provide the position and velocity of the oscillating mass.

Euler’s formula writes e⁻ⁱᵗ = cos t − i sin t. If position starts at 1 and velocity at 0, position follows cos t and velocity follows −sin t: two sinusoidal views of the same circular motion.

The imaginary-interest question is unrealistic for money, but extending the mathematics leads to a concrete description of oscillation. Sanderson points out that related complex-number formulations also appear in more advanced mechanics.

In the final audience question, he returns to a previous homework problem about matrix exponentials. The exponential can be defined by its power series for matrices as well as ordinary numbers.

The familiar identity exp(A + B) = exp(A)exp(B) depends on commutation in the usual proof. Matrices need not commute, so the identity is not generally valid for arbitrary matrices, though it holds when A and B commute.

Sanderson warns that exponent notation can suggest rules that no longer hold in noncommutative settings, including matrices and quaternions. The homework is meant to expose the role of commutativity in the binomial expansion.