Algebraic approach
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!
All articles in this folder
What is a complex number?
The set of complex numbers gained acceptance—with difficulty—when it became necessary to look beyond the real numbers for all the solutions of a quadratic equation. What no one had anticipated was its richness and the links it would forge with mathematics as a whole. Early discoveries.
? is an algebraically closed field
The field ? of complex numbers was constructed to provide solutions to every quadratic equation. Surprisingly, it also contains the solutions to all algebraic equations with coefficients in ?. In technical terms, it is algebraically closed.
Speeding up integer multiplication | Tangente
Complex numbers seem very far removed from the modern world’s concerns about profitability. Yet they underpin methods used to speed up the multiplication of large integers. They save time—a great deal of time—and therefore money!
Complex numbers of modulus 1
Initially mere formal symbols used in algebraic calculations, complex numbers came into widespread use from the 19th century onward thanks to their geometric interpretation... in almost every branch of mathematics! They therefore arise naturally in number theory.
Conjugates, moduli and arguments
Once we accept the existence of a number i such that i² = −1, do we risk losing touch with physical reality? Quite the opposite: a profound and fruitful correspondence emerges, allowing questions of pure geometry to be solved through simple algebraic manipulations.