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.
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Studying isometries of the plane—reflections, translations, rotations, and so on—can sometimes be dizzying. What happens to a point or figure when several transformations are applied in succession? Complex numbers provide a representation that is as elegant as it is illuminating.
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.
Complex numbers, born of impossible calculations, found an unlikely geometric interpretation. This meeting of algebra and geometry is beautifully illustrated by theorems about the roots of a complex polynomial.
Isometries (and more generally similarities) are not the only plane transformations that can be easily described using complex numbers. The same is true of inversion.
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