How Imaginary Numbers Were Invented

Mathematics originated as a tool for measurement and prediction. The separation of math from the real world led to the invention of imaginary numbers, which are now integral to our understanding of the universe.

Luca Pacioli, a math teacher in Renaissance Italy, concluded that finding a general solution to the cubic equation is impossible. Ancient mathematicians used visual methods and geometry to solve equations without written formulas. They struggled with negative solutions and coefficients due to their aversion to negative numbers.

The Secret of the Depressed Cubic In 1510, Scipione del Ferro solves a subset of cubic equations but keeps it secret to secure his job. After nearly two decades, he reveals the solution to his student Antonio Fior who boasts about it and challenges mathematician Niccolo Fontana Tartaglia in Venice. Tartaglia effortlessly solves all 30 problems given by Fior.

Tartaglia's Method and Triumph Niccolo Fontana Tartaglia, known as 'stutterer' due to a childhood injury, uses an innovative geometric approach involving completing the cube to solve depressed cubics. He extends the idea of completing the square into three dimensions and derives an algorithm presented as a poem for solving such equations.

The Secret of Tartaglia's Victory Tartaglia refuses to reveal his method for solving the cubic, but Cardano persists and lures him to Milan. After forcing Cardano to swear an oath not to disclose the method, Tartaglia reveals it. Cardano then discovers a solution for the full cubic equation and wants to publish it despite his oath.

Cardano's Method: The Geometric Paradox While writing 'Ars Magna,' Cardano encounters cubic equations with solutions containing square roots of negative numbers. He realizes that these negative areas are crucial intermediate steps in finding solutions. Bombelli later introduces a new type of number involving the square root of negative one, allowing mathematicians to find correct answers by abandoning geometric proofs.

The Emergence of Imaginary Numbers Rene Descartes popularized the use of square roots of negatives, calling them imaginary numbers. These new numbers liberated algebra from geometry and led to the development of complex numbers. Schrödinger's wave equation for quantum particles featured i, the square root of negative one, which physicists found uncomfortable in a fundamental theory. However, imaginary numbers have unique properties that make them essential in describing reality.

Complex Numbers and Quantum Physics Imaginary numbers exist on a dimension perpendicular to real number line and form the complex plane when combined with real numbers. Multiplying by i causes rotation by 90 degrees in the complex plane; this concept is utilized in e^ix function which contains cosine and sine waves - quintessential functions for describing waves. The solutions to Schrödinger's wave equation involve e^ikx-ωt due to its useful properties such as proportionality under derivative operations.