The saga of imaginary numbers – 05/05/2021 – Marcelo Viana

At the end of his book Summa, published in 1494 (six years before the arrival of the Portuguese in Brazil!), Luca Pacioli (1445–1517) wrote: “in the current state of science, solving the cubic equation is as impossible as the square of the circle”. But, within a decade, Scipione del Ferro (1465–1526) found a method of solving the cubic, which was soon generalized by Niccolò Tartaglia (1500–1557).

The problem was that, in many cases, the resolution involved square roots of negative numbers, which did not seem to make sense. In “L’algebra” (Algebra), published in 1572, Rafael Bombelli (1526–1572) explained how to operate with this new type of numbers to find all the solutions to any cubic equation.

But these figures continued to be viewed with suspicion, because they lacked physical interpretation. From that period, there remains the regrettable designation of “imaginary”, which suggests – wrongly – that such numbers would be less legitimate than the others. It goes back to “La géometrie” (Geometry), published in 1637 by the great French philosopher and mathematician René Descartes (1596–1650): “For each equation we can imagine as many solutions as their degree suggests, but in many cases the amount of solutions is less than we imagine ”.

Euler (1707–1783) introduced the symbol i to represent the square root √-1 of the number –1, which appears in his famous formula e+ 1 = 0. Gauss (1777–1855) was also interested in the numbers a + bi, which he called “complexes”. He began to point to the physical interpretation of these numbers, which would be given by Wessel and Argand.

In 1797, the Norwegian Caspar Wessel (1745–1818) proposed that, just as real numbers correspond to points on a line, as taught by Greek geometry, complex numbers are represented by vectors on the plane.

Written in Danish, Wessel’s work was forgotten for more than a century, losing credit for the French article published in 1806 by Jean-Robert Argand (1768-1822), with a similar proposal, which definitively resolved the question of legitimacy of complex numbers. Ironically, Argand almost lost his credit too, because he forgot to write his name in the article!

But the great rematch of complex numbers occurred as early as the 20th century, when quantum mechanics came to show that they are indispensable to describe the physical universe. It is not every day that we, mathematicians, discover something and leave the physicist colleagues the task of verifying that it is right!

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