Complex numbers

Irrational numbers, zero, and negative numbers took centuries to be accepted. This was also the case for the "imaginary" numbers, which gave rise to the notion of complex numbers. At the origin of their – late – introduction, there was the wish to solve quadratic equations that had no real solution. This led to the conception of a powerful set, possessing the structure of an algebraically closed field, that is, in which every algebraic equation admits a solution. Better, since complexes can be identified to a point in the plane, a powerful correspondence between algebra, analysis, geometry and trigonometry was to be born!

First things first: geometry is the first to benefit from the introduction of imaginary numbers. The representation of complex numbers as points in the plane allows one to cleverly 'encode' a transformation, to judiciously 'capture' the locus of a moving point. Homotheties, similarities and other inversions thus receive a simple algebraic interpretation and become easily manipulable. Thanks to the powerful tool of complex numbers, geometric results can be demonstrated, or even be discovered, such as Marden's theorem. Concepts, like that of fractals, can be highlighted.

Complex numbers totally revolutionized analysis: by allowing the variable of a humble real function to take values in ℂ, Leonhard Euler and especially Bernhard Riemann opened a Pandora's box whose richness no one could have imagined. The exponential finally flourished, and with it all of trigonometry, whose formulas become accessible to everyone! Such that domains of physics, like electrical engineering, can no longer do without it. The zeta function now makes us gaze at a thousand mathematical wonders, especially regarding prime numbers, the elementary building blocks of arithmetic. But beware the rash soul who ventures into the Riemann hypothesis: the million-dollar reward promised for its proof alone speaks to the immense scale and formidable challenge of the task…