What do complex numbers really represent? How can we "picture" i² being equal to −1? A striking visual answer comes from interpreting multiplication geometrically. The icing on the cake is that the same model explains why "a negative times a negative makes a positive."
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René Descartes dreamed of turning a problem in pure geometry into an algebraic one. It took more than a century to realize that dream! Complex numbers opened up a new way to explore geometric figures and constructions.
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.
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.
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.
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