Imaginary Numbers – 04/27/2021 – Marcelo Viana

“Artis Magnae” (The great art), published in 1545 by the Italian Gerolamo Cardano (1501–1576), is one of the three most important scientific works of the Renaissance of the Celestial Spheres) by Nicolau Copérnico, along with “De revolutionibus orbium coelestium” (Of the Revolutions) and “De humani corporis fabrica” ​​(On the organization of the human body) by Andreas Vesulius, both published in 1543.

Cardano discusses, among other things, the solution of the special cubic equation x3 + mx + n = 0, which was discovered some time before by his compatriot Scipione del Ferro.

It was already known that the quadratic equation ax2 + bx + c = 0 can have two solutions, one or neither, depending on the value of the “discriminant” b2-4ac. If the discriminant is negative, then the methods of solving (for example, the “Baskhara” formula) include computing square roots of negative numbers, which at that point were considered meaningless expressions, and then the equation has no solution.

The cubic equation x3 + mx + n = 0 behaves similarly, but is much more interesting. Its discriminant is m2 / 4 + n3 / 27 and when it is negative, del Ferro’s formula also includes square roots of negative numbers. However, in this case there can be up to three solutions. This is the maximum number for a class 3 equation.

How can the formula contain meaningless expressions even if there are legitimate solutions ?! Although Cardano did not understand, he said that “we should be calculating anyway” and treated such expressions as if they really were numbers. But he could neither apply this idea to the cubic equation nor understand its usefulness.

It was also the Italian Rafael Bombelli (1526–1572) who developed the theory of so-called imaginary numbers. In his work “L’algebra” (Algebra) (in Italian, not Latin, which was a great novelty) published in 1572, Bombelli called the square root √-1 of the number -1 “more than less” and explained how to operate on it perform a new type of number (“more of less times more than less equals less”).

Bombelli applied his theory to the cubic equation, explaining how, in the situation Cardano had fascinated, it is possible to get all three “real” solutions to the equation using del Ferro’s formula using imaginary numbers. It was a case of ends (solving the equation) that justified the means, in this case the use of square roots of negative numbers which “must seem like a cheat to most people”.

This dubious situation regarding imaginary numbers would continue into the 19th century.

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