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The classics of mathematical humor often borrow some of their gems from the vocabulary of complex numbers…


The classics of mathematical humor often borrow some of their gems from the vocabulary of complex numbers…


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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."

How can the classical exponential function be extended to complex numbers? Will its usual properties be preserved? Although the resulting extension is easy to study, the associated notion of a complex logarithm is more elusive. It was the subject of controversy in the 18th century.

Complex numbers, initially termed "imaginary," were not conceived as we study them today. Above all, they were introduced as tools for solving polynomial equations—and tackling the mathematical challenges that raged across Renaissance Europe.

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.
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