COMPLEX NUMBER. 236 COMPLEX NUMBER. is commonly called the imaginary unit, and is represented " by i. All numbers containing the factor I are called imaginary numbers, as op- posed tq, real numbers ; e.g. ± i, ±z '21, =b3(, . . . . ± I », ± V2 . i are imaginaries. The algebraic sum of a real number and an imaginary is called a complex number: e.g. 1 + i. 2 — 4i, and in general a + bi. A complex variable is generally ex- pressed by X -!- yi, in which x and y are real Y ^h ."3 11
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X -3 -2 1 1 -i -a -3i V' 2 3 variables. Complex numbers are represented graphically in a plane. In the figure the real numbers are laid off on the axis XX' in the usual way, and the coefficients of i on the axis YY'. The points in the plane corresponding to these coordinates represent the complex numbers. Tnus. Pi on the axis represents the real number 2, r~ represents the comijlex number 3 + 2i, P, repre- sents 3i, and Pj represents — 1 — 3(. Any point and the origin uniquely determine a line-segment, or vector, called the modulus of the complex num- ber, and this may also be taken to represent the number. In the figure, the moduli are OP,, OP;, OP;, OP,. In general, the modulus of a com- plex number n -f- bi is the diagonal of a rec- tangle of sides « and b ; hence its absolute value is ^/a' -- b". Thus, the modulus of 3 -f 2 i (OP, in the figure) is /9 + 4 or l/ 13. The convention as to the direction of i is a reasonable one : for since multiplying + 1 by — 1 revolves it through ISO" to the position — 1, therefore its multiplication by one of the two equal factors of — 1, viz. i/ — 1, may be interpreted as revolving it through 90°. There are other sufficient reasons for this convention, which will be evident to one who studies the subject. The complex number is a directed mag- nitude; that is, it has both extension and direc- tion in its plane. This is best understood by considering n -i- bi in the form rlcosS -- i sin 6), in ■which r is the modulus -^ cr -f W, and 8 is the amplitude. In the figure, cos = 6 Two complex numbers which dift'or only in the sign of the imaginary part are called conjugates; e.g. 2 -y 'Ai and 2 — 3(, or. in general. « + bi and a — bi. Complex numbers are subject to the associative, commutative, and distributive laws, and, when combined l)y the fundamental operations of algebra, yield im number not already defined. For x -f- yi represents real numbeis ^^•hen y = 0. iniaginaries when a; = 0, and complex numbers when x, y are real and not zero. Hence. ,r -f- yi becomes a convenient form for representing general numbers : and instead of saying that every equation has a root, which may be real, imaginary, or complex, we may say that every equation has a root x -- yi. If, in plotting the successive moduli of a sum, the second modulus is drawn from the end of the first, the third from the end of the second, and so on, the result is a broken line which may be closed by connecting the last point with the origin. This vector is called the sum. Since no side of a polygon is greater than the sum of the rema.ining sides, the modulus of the sum of any number of complex numbers is not greater than the sum of their moduli. This is expressed symbolically thus : |Nn|<|N,| + |N,|+....|Nn-,| -Multiplying r (cosS -f inme) by r' (cosB' + i sine') and applying the formulas for the func- tions of the sxim of two angles (see Trigonom- etey), the product is (•/■'[coslff -|- $') + i sin I 8 -|- 9')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 h com- plex numbers. For brevity, let /-cisS = r(cos9 -h i sine), then r, cis fl, • r,eise; ' . . rncis ^n = i',' r, • ... rncis (^, + e, -+-... Sn I. This is known as De iloivre's theorem. If each of the above num- bers equals the first, (r,cise,) ii:= r, n cis nS, or the (itli power of the complex number. Th'e quotient of r,ci?e, by r»cise, =. — cis(f?i — d^). V(i' b- , sin 9 = Va- + b'- = ( See Trigonometry. ) This method of representing the complex num- ber as a directed magnitude in a plane was at one time thoight to he due to Argand. and hence the figure is often called Argand's diagram. and {//■, cisSj = -p/ r, • eia — f). 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 lias a root is a consequence of Cauchy'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 Stereometria of Hero of .lexandria (third century B.C.). Diophantus (supposed to have llourislied in the fourth century A.D. ) met these numbers in his algebraic work, but failed to give an explanation. Bhaskara (.^.n. 1114) recognizes the imaginary, but pronounces the roots involv- ing V — 1 to be impossible. Cardan (1.545), in his A rs jrafina. was the first mathematician who had the courage to use the square roots of negative numbers in computation. Bombelli, Giiard, and Descartes (q.v.) formulated rules
