## 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: “According to the current state of science, the solution of the cubic equation is as impossible as possible the square of the circle” Within a decade, Scipione del Ferro (1465–1526) found a method for 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 didn’t make sense. In “L’algebra” (Algebra) published in 1572, Rafael Bombelli (1526–1572) explained how to work with this new type of number to find all the solutions to any cubic equation.

However, these numbers continued to be viewed with suspicion as they could not be physically interpreted. From this period the unfortunate term “imaginary” remains, which wrongly suggests 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 think of as many solutions as its degree suggests, but in many cases those The number of solutions is fewer 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 eiπ + 1 = 0. Gauss (1777–1855) was also interested in the numbers a + bi, which he called “complexes”. He began to point out the physical interpretation of these numbers that Wessel and Argand would give.

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

Wessel’s work, written in Danish, had been forgotten for more than a century and lost recognition for the French article published in 1806 by Jean-Robert Argand (1768-1822) with a similar proposal addressing the question of the legitimacy of complex numbers finally resolved. Ironically, Argand almost lost his credit because he forgot to put his name on the article!

However, the great rematch of complex numbers took place as early as the 20th century when quantum mechanics showed that they are essential for describing the physical universe. It is not every day that we mathematicians discover something and leave the task of our physicists to check whether it is correct!

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