502 EQUATION solution of a simple equation leads to an extension ; ax - b = 0, gives x , a positive fraction, and we can in CL this manner represent, not accurately, but as nearly as we please, any positive magnitude whatever; so an equation = Q gives x= - - , which (approximately as before) represents any negative magnitude. We thus arrive at the extended signification of number as a continuously varying positive or negative magnitude. Such numbers may be added or subtracted, multiplied or divided one by another, and the result is always a number. Now from a quadric equation we derive, in like manner, the notion of a complex or imaginary number such as is spoken of above. The equation # 2 +l=0 is not (in the foregoing sense, number - real number) satisfied by any numerical value whatever of x ; but we assume that there is a number which we call i, satisfying the equation i 2 + 1 = ; and then taking a and b any real numbers, we form an expression such as a + bi, and use the expression number in this extended sense : any two such numbers may be added or subtracted, multiplied or divided one by the other, and the result is always a number. And if we consider first a quadric equation x z +px + q = where p and q are real numbers, and next the like equation, where p and q are any numbers whatever, it can be shown that there exists for x a numerical value which satisfies the equation ; or, in other words, it can be shown that the equation has a numerical root. The like theorem, in fact, holds good for an equation of any order whatever ; but suppose for a moment that this was not the case ; say that there was a cubic equation x 3 + px 2 + qx + r = 0, with numerical coeffi cients, not satisfied by any numerical value of x, we should have to establish a new imaginary j satisfying some such equation, and should then have to consider numbers of the form a + bj, or perhaps a + bj + cj 2 (a, b, c numbers a-f j3i of the kind heretofore considered), first we should be thrown back on the quadric equation x- + px + q = 0, p and q being now numbers of the last-mentioned extended form non constat that every such equation has a numerical root and if not, we might be led to other imaginaries k, I, &c., and so on ad infinilum in inextricable confusion. But in fact a numerical equation of any order whatever has always a numerical root, and thus numbers (in the fore going sense, number = quantity of the form a + /3i) form (what real numbers do not) a universe complete in itself, such that starting in it we are never led out of it. There may very well be, and perhaps are, numbers in a more general sense of the term (quaternions are not a case in point, as the ordinary laws of combination are not adhered to), but in order to have to do with such numbers (if any) we must start with them. 1C. The capital theorem as regards numerical equations thus is, every numerical equation has a numerical root ; or for shortness (the meaning being as before), every equation has a root. Of course the theorem is the reverse of self- evident, arid it requires proof ; but provisionally assuming it as true, we derive from it the general theory of numerical equations. As the term root was introduced in the course of an explanation, it will be convenient to give here the formal definition. A number a such that substituted for x it makes the function x-? jo^"" 1 . . . p n to be = 0, or say such that it satisfies the equation f(x) 0, is said to be a root of the equation; that is, a being a root, we have and it is then easily shown that x -a is a factor of the f unction /(j-), viz., that we have/(.r) = (x a) f-^x), where fi(x) is a function x"" 1 - q ] x n ~ 1 . . . q n _ 1 of the order n - 1, with numerical coefficients q v g 2 . . </_! In general a is not a root of the equation f-^x) = 0, but it may be so i.e., f^(x) may contain the factor x-a; when this is so, /(./;) will contain the factor (x - a) 2 ; writing then/() = (x - a) 2 f 2 (x), and assuming that a is not a root of the equation fr(x) = 0, x = a is then said to be a double root of the equation f(x) = ; and similarly f(x) may con tain the factor (x - a) 3 and no higher power, and x = a is then a triple root ; and so on. Supposing in general that/(x) = (x - a) a F(.r) (a being a positive integer which may be = 1, (x a) a the highest power of x-a which divides f(x), and F(x) being of course of the order n - a), then the equation T?(x) = will have a root b which will be different from a; x-b will be a factor, in general a simple one, but it may be a multiple one, of F(x), and f(x) will in this case be = (x - a) a (x - b)& <b(x) (fi a positive integer which may be = 1, (x- b)& the highest power of x-b in F(#) or f(x), and <(.*) being of course of the order n - a - (3)._ The original equation f(x) = is in this case said to have a roots each a, (3 roots each =b ; and so on for any other factors (x - e)?, &c. We have thus the theorem A numerical equation of the order n has in every case n roots, viz., there exist n num bers a, b, . . (in general all distinct, but which may ar range themselves in any sets of equal values), such that f(x) = (x- a)(x - V)(x - c) . . . identically. If the equation has equal roots, these can in general be determined, and the case is at any rate a special one which may be in the first instance excluded from considera tion. It is, therefore, in general assumed that the equa tion f(x) - has all its roots unequal. If the coefficients p v p 2 . . . are all or any one or more of them imaginary, then the equation f(x) = 0, sepa rating the real and imaginary parts thereof, may be written F(x) + i$>(.c) - 0, where F(x), 4>(x) are each of them a function with real coefficients; and it thus appears that the equation f(x) 0, with imaginary coefficients, has not in general any real root ; supposing it to have a real root a, this must be at once a root of each of the equa tions Y(x) = and *(*) = 0. But an equation with real coefficients may have as well imaginary as real roots, and we have further the theorem that for any such equation the imaginary roots enter in pairs, viz., a + (3i being a root, then a /3i will be also a root. It follows that if the order be odd, there is always an odd number of real roots, and therefore at least one real root. 17, In the case of an equation with real coefficients, the question of the existence of real roots, and of their separa tion, has been already considered. In the general case of an equation with imaginary (it may be real) coefficients, the like question arises as to the situation of the (real or imaginary) roots; thus, if for facility of conception we re gard the constituents a, ft of a root a + /?i as the coordi nates of a point in piano, and accordingly represent the root by such point, then drawing in the plane any closed curve or " contour," the question is how many roots lie within such contour. This is solved theoretically by means of a theorem of Cauchy s (1837), viz., writing in the original equation x + iy in place of x, the function f(x + iy) becomes = P + z Q, where P and Q are each of them a rational and integral function (with real coefficients) of (x, y}. Imagining the point (x, y) to travel along the contour, and considering the number of changes of sign from - to + and from + to - of the fraction corresponding to passages of the fraction through zero (that is, to values for which P becomes = 0, disregarding those for which Q becomes = 0), the difference of these numbers gives the number of roots within the
contour.