e, i, and the Long Detour
Into Quantum Mechanics

Two ideas born from unglamorous problems, compound interest and an unsolved cubic equation, turned out four centuries later to be the native language of quantum theory. Neither Cardano nor Bernoulli had any idea what they'd started.

1545

Cardano publishes the cubic formula

Ars Magna solves cubic equations algebraically. For some cubics with three perfectly real roots, the formula demands taking the square root of a negative number along the way.

1572

Bombelli does the forbidden arithmetic

Rather than reject the square root of −1, Bombelli treats it as a formal object, runs the algebra, and watches the imaginary parts cancel to leave the correct real answer.

1614

Napier introduces logarithms

A tool for turning multiplication into addition, and the seed of the exponential relationships Euler would later unify.

1637

Descartes coins "imaginary"

Meant dismissively, in La Géométrie, for numbers he considered fictitious. The name stuck long after the dismissal didn't.

1683

Bernoulli finds e hiding in compound interest

Asking what happens to interest compounded infinitely often, Jacob Bernoulli lands on the limit (1+1/n)ⁿ, without yet naming it.

1748

Euler unifies growth and rotation

Introductio in analysin infinitorum gives the world e^iθ = cos θ + i sin θ, and its famous special case e^iπ + 1 = 0.

1799

Complex numbers get a home

Gauss, and independently Argand and Wessel, plot complex numbers on a plane. Multiplying by i stops being mysterious: it's a quarter turn.

1822

Fourier decomposes arbitrary signals

Théorie analytique de la chaleur shows any periodic function is a sum of sines and cosines, Euler's rotating exponentials in disguise.

1926

Schrödinger writes i into physical law

The wave equation for quantum mechanics has i built into its core. The "imaginary" detour of 1545 becomes literally load-bearing.

e — the number of continuous growth

Put $1 in an account paying 100% annual interest. Compound it once a year and you get $2. Compound it monthly, daily, by the second, and the total keeps creeping up, but not without limit. It settles on one specific number.

e = lim_n→∞ (1 + 1/n)ⁿ ≈ 2.71828…

What actually makes e important isn't the interest problem, it's that e^x is the one function whose rate of change equals its own value. Anything that grows or decays in proportion to how much of it there already is, populations, radioactive decay, capacitor voltage, is secretly writing itself in terms of e.

i — the detour nobody trusted

x³ − 15x − 4 = 0, three real roots, marked on the curve

This cubic has three ordinary, real, solutions: 4, and −2±√3. No mystery there. But Cardano's own formula for solving cubics, applied to this equation, produces an intermediate step where you must take the square root of −121.

The answer at the end is completely real. The path to it isn't. Bombelli's move, in 1572, was to stop refusing that detour and just carry the bookkeeping through: let i mean √−1, follow the algebra, and trust that it would resolve itself. It did. That's the first time anyone treated imaginary numbers as usable rather than nonsensical.

i as a quarter turn

repeated multiplication by i: 1 → i → −1 → −i → 1

The geometric picture, arriving two and a half centuries after Bombelli, is what finally made i feel like a real object instead of a trick. Multiplying a number by i doesn't stretch or shrink it, it rotates it 90° on the complex plane.

i·1 = i i·i = −1 i·(−1) = −i i·(−i) = 1

Four multiplications, one full turn. That's the entire content of "imaginary": a name for the operation of turning.

Euler's identity: growth meets rotation

e^x is defined by an infinite sum: 1 + x + x²/2! + x³/3! + … Feed in x = iθ instead of a real number, and each term becomes a little arrow in the complex plane, its length shrinking fast thanks to the factorial, its direction cycling through the four quarter-turns from panel 03.

e^iθ = 1 + iθ − θ²/2! − iθ³/3! + θ⁴/4! + …

Chain those arrows tip to tail and watch where they land. At θ = π they walk, term by term, straight to the point −1.

e^iπ + 1 = 0

Growth (e), rotation (i), and the circle (π) turn out to be one and the same idea, viewed from three different centuries of mathematics.