COMPLEX NUMBER. The steps in the growth of the system of algebra may eas ily he illustrated by the roots of equations. thus: The solution of the equation s 3 = (1 is 3. a positive integer which may he represented graphically on a straight line. The solution of the equation 3.r-2 = 0 is '4,', a fraction which may also be represented graphically on a straight line. The sol ttm a of 2 = 0 is 1 '2, a surd which may he represented by the diagonal of a square whose side 1. The solution of 2 = (1 is 2. a negative number, which may he rep resented on a straight line in the opposite direc tion from that of the positive number. But the solution of a.= 0 =0 is ± or called an imaginary number. The symbol 11 is commonly called the imaginary unit, and is represented by i. All numbers containing the factor i are ealled imaginary numbers, as op posed to real numbers: e.g.
± i, 2i. -,--3i, are imaginaries. The algebraic sum of a real number and imaginary is called a complex number: e.g. 1 i. 2 4i. and in general a --I- 61. A complex variable is generally pressed by x yi, in which a; and y arc real 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 X Y'. The points in the plane corresponding to these coordinates represent the complex numbers. Taus. I', on the axis represents the real number 2, P, represents the complex number 3 4- 21. P, repre sents 3i, and P, represents I 3i. Any point and the origin uniquely determine a line-segment, or vector, called the modulus of the complex num ber, nail this may also he taken to represent the number. In the ficuire, the moduli are OP,. OP,, OP . OP,. In the modulus of a com plex number a hi is the diagonal of a rec tangle of sides e and ; hence its absolute value is 1 a= ± Thus, the modulus of 3 + 21 (OP, in the figure) is 1/9 + 4 or 1 13. The convention as to the direction of is a reasonable one: for since multiplying + by 1 revolves it through ISO' to the position 1. therefore its multiplication by one of the
two equal factors of 1. viz. 1/ 1. may he interpreted as revolving it through 90'. .There are other sufficient reasons for this convention. which will he evident to one who studies the subject. The complex nuniher is a directed mag nitude: that is, it has both extension and direc tion in its plane. This is best understood by considering a 12i in the form r (er1.0 i sin e), in which r is the modulus and e is the amplitude. In the figure, cos e = , , 0 = (See TRIGONOMETRY.) b= 1' + b= This method of representing the complex num ber as a directed magnitude in a plane was at one time thought to be due to Argand, and hence the figure is often called Argand's diagram.
Two complex numbers which differ only in the sign of the imaginary part are called conjugates; e.g. 2 + 3i and 2 3i, or. in general. b and a bi. Complex numbers are subject to the associative, commutative, and distributive laws, and, when combined by the fundamental operations of algebra, yield no number not already defined. For .r yi represents real when a = 0. imaginaries when a' = 0, and complex numbers when .r, y are real and not zero. Hence. ,r + 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 + pi. f, 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 remaining 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: !Nei ± Judtiplying r isine) by r'(cosO' i and applying the formulas for the func tions of the sum of two angles (see TRIGONOM