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MathHistory15: Complex numbers and algebra

Complex numbers of the form a+bi are most introduced these days in the context of quadratic equations, but according to Stillwell, cubic equations are closer to their historical roots. We show how the cubic equation formula of del Ferro, Tartaglia and Cardano requires some understanding of complex numbers even when only real zeroes appear to be involved. The use of imaginary numbers in calculus manipulations is illustrated with some computations of Johann Bernoulli relating the inverse tan function to complex logarithms, and the connections bewteen tan (na) to tan(a). The geometrical planar representation of complex numbers goes back to Cotes, Euler and DeMoivre in some form, and then more explicity at the end of the 18th century by Wessel and Argand, and then by Gauss. The Fundamental theorem of algebra is a key undergraduate result that often proves elusive---it was so also for the pioneers of the subject. Euler, Gauss and d'Alembert all struggled with the result, but made progress. Here we outline the ideas behind the proofs of d'Alembert and Gauss.
Length: 01:07:16


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