ETRY), the product is rr'reos( C + 0') i in + 1. Hence, the product of the moduli of two complex numbers is the modulus of their product. and the sum of the amplitudes is the amplitude of the product. Similarly for n com plex numbers. For brevity, let r(cos0 i sine), then r, cis r,cise, • .. = r,• r, • ... 8, ... This is known as De Xloivre's theorem. If each of the above num bers equals the first, (r,eis cis or the nth power of the complex number. Th'e miotient of r,ci4 0, by r,cise, = 7 eis and else, ' r, • cis By observing the changes in the modulus and amplitude. the results of any of these operations may he represented graphically. The variation of a function of a complex variable, x yi, due to the variation of x and y, is very important in the theory of equations and functions. Thus the fundamental proposition that every equation has a. root is a consequence of Canchy's theorem which asserts that the number of roots of any equation comprised within a given plane area is obtained by dividing by 2- the total variation of the amplitude of the function corresponding to the complete description, by the complex vari able. of the perimeter of the area.
The first appearance of the imaginary is found in the Strreometrin of Hero of Alexandria (third century n.c.). Diophnntus (supposed to have flourished in the fourth century A.D.) met these numbers in his algebraic work, but failed to give an explanation. Bhaskara (A.D. 1114) recogDizes the imaginary, hut pronounces the roots invol• in 1/-1 to be impossible. Cardan (1545), in his A rs Magna. was the first mathematician who had the courage to use the square roots of negative numbers in computation. Bombelli, Girard, and Descartes (q.v.) formulated rules for the use of such quantities as a +LI but founded no theory. Wallis (1635) made the first attempt to give a geometric interpreta tion. Euler (1770) still regarded these quanti ties impossible. Thus it was reserved for Casper
Wessel (1797), a Norwegian surveyor, to invent a graphic treatment of complex Ilis method is contained in a memoir, presented to the Royal Academy of Science and Letters of Denmark. entitled On the .1nalatic Representa tion of Direction. For the early development of the subject, however, credit must be given to Argand. Gauss. Servois. and others, since Wes !sel's article (published in 1799 by the !loyal Academy of Denmark) dial not appear in French until 1397, one hundred years after its presen tation. Gauss did much to establish the under lying principles. Argand's memoir (1300. un quest i(mahly an original and independent pro duction, supplied the graphic theory that lay neglected in the work of Wessel. Francois, Ser vois, Geirnme, and Cauchy did much to correct the errors of their predecessors and to generalize the theory of directed lines.
Complex number. being the most general type of algebraic number, has come to occupy the place of highest importance in modern analysis. It has led in recent times to the establishment of the theory of functions (q.v.) and quaternions (q.v.). Consult: Berman, "A Chapter in the His tory of Mathematics," in the Proceedings of the American A ssocia t ion for the Jdrancement of ticlence (Salem, 13971 ; Cauchy, ('ours d'amtlyse (Paris, 1321) : Warren, _I Treatise on the Geo metric Representation of the Srprare Roots of 1Vegatire Quantities (Cambridge, 13271: Chrys tal, .11gebra, part i. (Edinburgh, 1839) ; Eankel, Forlesungcn ii her die romplexen Zahlrn c Leip zig, 1307) Durege, Theorie der Functioncn ricer eomple?en reriinderli!ehen Greisse (Leip zig. 1373), trans. by Fisher and Sehwatt as Ele ments of the Theorp of Functions of a Complex Variable (Philadelphia, 1396).