The following equations can not be decomposed rationally into quadratic factors, hence the resolvent cubic does not have any rational root. The cubic equation must be solved by the method of Cardan or by the trigonometric method. CHAPTER IV THEN ROOTS OF UNITY 779. Given the equation cos'n A+ i sin n A = (cos A + i sin A)", _i = √ −1, it is proposed to determine the values of n A from the sine and cosine of n A. It will be found that there are n different values of A which give different values of cos Aisin A, and which equally satisfy the equation. For, by Trigonometry, we know that the terms of the two series and no others have the same cosines; hence it follows that the terms of the series n A, 2π+nA, 4′′ +n A, 2 (n−1)+ n A, 2 n π + n A, and no others have simultaneously the same sines and cosines, and hence each gives the same value for the expression cos n Aisin n A when both cosine nA and sine nA are given. Therefore the n terms of the series A, +A, + A, 2π Απ n 2(n-1) T +A, n and no others, give different values for cosine A and sine A (one or both), and hence different values for cos Aisin A; but they give the same values to both terms of the expression cos n Aisin n A. Therefore n values of this series are the only values of A which satisfy the equation (cos Aisin A)" cos nA + i sin n A. = Hence it follows that any one of the n expressions raised to the nth power will give the same value for the expression cos n Aisin n A, provided n A is to be determined from the value of its sine and cosine. NOTE. Since sin (2π- x) = - sin x and cos (2π- x) = cos x, it follows that and therefore that the n different values in the preceding table can be reduced to equivalent expressions, which involve no angle greater than 180°. 780. Solution of the Equation x"=1.—The roots of the equation x2-1=0 may readily be found from the equation (1) [cos(+4)+ i sin (2n+1)]" = cos n A+ i sin n A, where, according to the preceding table, p=0, 1, 2, . . . . n-1. Put A = 0, and hence cos n A=1, sin n A=0, and we have 781. The Cube Roots of Unity.- Let n = 3 and 40, then it follows from (2) and table in 8779 that the cube roots of unity are expressed by These results agree with those already given in 2767. 782. The Biquadratic Roots of Unity. Let n = 4 in (2), 2780, and table in 2779, and the four roots are expressed by 783. The Quinary Roots of 1.- -Let n=5 in (2), 780, and table in 2779, and the required roots are 784. The Correspondence Between the Complex Number System and the Points of a Plane. All the numbers included in the system of complex numbers a+ib can be represented by the points in a plane. A' N B Y r P a M JA FIGURE 1 X Let XOX and YOY' be two perpendicular lines lying in the same plane and intersecting in 0. Suppose that XOX' is the axis of real numbers and let the real numbers be represented by the points of XOX' as described in 231, and suppose that YOY' is the axis of pure imaginary numbers, representing ib by the point of OY whose distance from O is b in case is positive, and at the same distance from O on OY' when bis negative. To construct the point corresponding to the complex number a + ib, where a and b are positive, lay off on OX, OM = a, and on the perpendicular to OX at M, MP = b. There is a one-to-one correspondence between the numbers of the complex system (a+b) and the points of the plane. To every complex number there corresponds one and but one point in the plane, and to each point of the plane there corresponds one and but one complex number. If the point P is made to move about in the plane OM, and PM varies, then aib varies and is called a complex variable, usually written in this case x + iy. 785. Modulus. The length of the line OP, which is Va2 + b2, is called the modulus of a + ib. Represent it by r, i. e., 786. Argument. The argument of the number a + ib is the positive angle XOP. Represent its numerical value by A. The angle A is always measured in the positive sense from XO toward the modular line OP. 787. Sine. The ratio of PA, the perpendicular from any point Pin the modular line to the axis of real numbers, to the distance of P from O is called the sine of A and is written sin A, i. e., Sin A is by definition positive when P lies above the axis of real numbers, and negative when Plies below. |