Uses Of Complex Numbers
Complex numbers are needed for the fundamental theorem of algebra: Every polynomial equation, P(x) = 0, with complex coefficients has a complex root. For example, the polynomial equation x2 + 1 = 0 has no real roots. If one allows complex numbers, however, it has two: 0 + i and - i.
Complex numbers are also used in the branch of mathematics known as "functions of complex variables." Such functions can often describe physical situations, as in electrical engineering and fluid dynamics, which real-valued functions cannot.
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J. Paul Moulton