Geometric representations
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.
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Complex numbers aren't so complicated
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."
Isometries of the plane
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.
Some interesting similarities...
Isometries can be studied very neatly using complex numbers. But they are not the only transformations that can be described by simple formulas! Once we dispense with preserving lengths, the vast family of similarity transformations opens up before us.
Another transformation: inversion
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.
The emergence of the complex plane | Tangente
The representation of the set of complex numbers by a plane appeared well after their invention.
The geometry of complex numbers
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.
Marden's theorem
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.