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HISTORY OF MATHEMATICS – LECTURE 18 – WEDNESDAY 17 TH NOVEMBER Complex Numbers Another problem area of mathematics where great progress was in the 19th century was that of complex numbers. The acceptance of complex numbers began with the derivation of the cubic equation by Cardano (see Lecture 8). However Cardano made little progress, noting that arithmetic that deals with quantities such as √െ1 "involves mental tortures and is truly sophisticated." At another point he concluded that the process is "as refined as it is useless." It was left to one of Cardano’s contemporaries, Rafael Bombelli (1526 1572), to see that understanding complex numbers was necessary to solving cubic and quartic equations. Ex. (from Bombelli’s treatise l’Algebra )
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While Euler advanced the theory of complex numbers by incorporating them into analysis, and simplifying calculations by writing i instead of root 1, there were many who would not be satisfied with them until they could be explained geometrically. This was looked at by Caspar Wessel (1745 1818) and Jean Robert Argand (1768 1822), who represented imaginary numbers as vectors in the x y plane. However Wessel and Argand were not regarded as great mathematicians, and so it was Gauss and his great reputation that finally allowed imaginary numbers (which Gauss referred to as “complex numbers” to remove some of the mystique) to become accepted more broadly by mathematicians. Instead of using vectors to denote the number a + bi, he used a point (a,b), and commented in an 1831 paper that “[Imaginary numbers] have hitherto been considered from the wrong point of view and surrounded by a mysterious obscurity. If for instance +1, 1, root 1 had been called direct, inverse, and lateral units, instead of positive, negative, and imaginary (or even impossible) such an obscurity would have been out of the question.” Both Cauchy and Riemann made further advances, deriving the main results that are seen in a complex variables class today, so that by the end of the 19th century the existence and theory of complex numbers was generally beyond question. However resistance did still persist, especially in England. Recalling his time as a student at Cambridge in the 1880‘s, A.R. Forsyth (1858 1942) wrote that “It was an age when the use of root 1 was suspect, even in trigonometric formulae. The imaginary i was suspiciously regarded as an untrustworthy intruder.” Given how Gauss represented the complex number a + bi by the point (a,b) in the plane, we can write their properties in purely algebraic terms, analogous to the way that was done for real numbers. For example: Addition: (a,b) + (c,d) = (a+c, b+d) Multiplication: (a,b) x (c,d) = (ac bd, ad +bc) Additive identity: (a,b) + (0,0) = (a,b) Multiplicative identity: (a,b) x (1,0) = (a,b) Commutative property for addition: (a,b) + (c,d) = (c,d) + (a,b) Commutative property for multiplication: (a,b) x (c,d) = (c,d) x (a,b)
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