After Bombelli, complex numbers long remained suspect computational tools: they appeared in formulas and gave correct results, yet no one could say what they were. Euler himself used them masterfully while still calling them “impossible”. The decisive turn was not a new calculation but an image: representing as the point with coordinates in the plane. In this way the imaginary unit ceases to be a mysterious symbol and becomes a precise direction — the rotation by a right angle with respect to the real axis.
Three almost independent discoveries
The idea of the complex plane was found several times, almost independently. The first was the Norwegian-Danish Caspar Wessel, who in 1799 presented to the Danish Academy a memoir in which complex numbers were vectors of the plane and multiplication corresponded to adding the angles and multiplying the lengths: a work that remained almost unnoticed for a century. In 1806 the Swiss Jean-Robert Argand, a Parisian bookseller and amateur mathematician, published anonymously the same geometric scheme, which today bears his name (Boyer; Katz).
Gauss and full legitimacy
The consecration came with Carl Friedrich Gauss, who had possessed the geometric representation since his youth and used it in his proofs of the fundamental theorem of algebra. It was Gauss, in a memoir of 1831, who proposed the term complex number in place of “imaginary” and publicly defended the full dignity of these numbers, stripping them of any air of paradox (Stillwell). Since then, writing and seeing it as a point of the plane are one and the same: the geometry of the Argand-Gauss plane is the natural language with which, in this chapter, we shall interpret modulus, argument, and conjugate.
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Topics: Complex numbers Concepts: Complex number · Argand-Gauss plane People: Caspar Wessel · Argand · Carl Friedrich Gauss