be a double root of the equation ƒ(x) = 0; and similarly ƒ(x) may contain the factor (x − a)3 and no higher power, and x = a is then a triple root; and so on.
Supposing in general that ƒ(x) = (x − a)αF(x) (α being a positive integer which may be = 1, (x − a)α the highest power of x − a which divides ƒ(x), and F(x) being of course of the order n − α), then the equation F(x) = 0 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 ƒ(x) will in this case be = (x − a)α (x − b)β Φ(x) (β a positive integer which may be = 1, (x − b)β the highest power of x − b in F(x) or ƒ(x), and Φ(x) being of course of the order n − α − β). The original equation ƒ(x) = 0 is in this case said to have α roots each = a, β roots each = b; and so on for any other factors (x − c)γ, &c.
We have thus the theorem—A numerical equation of the order n has in every case n roots, viz. there exist n numbers, a, b, ... (in general all distinct, but which may arrange themselves in any sets of equal values), such that ƒ(x) = (x − a)(x − b)(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 consideration. It is, therefore, in general assumed that the equation ƒ(x) = 0 has all its roots unequal.
If the coefficients p1, p2, ... are all or any one or more of them imaginary, then the equation ƒ(x) = 0, separating the real and imaginary parts thereof, may be written F(x) + iΦ(x) = 0, where F(x), Φ(x) are each of them a function with real coefficients; and it thus appears that the equation ƒ(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 equations F(x) = 0 and Φ(x) = 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. α + βi being a root, then α − βi 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.
9. In the case of an equation with real coefficients, the question of the existence of real roots, and of their separation, 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 regard the constituents α, β of a root α + βi as the co-ordinates of a point in plano, 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 A. L. Cauchy (1837), viz. writing in the original equation x + iy in place of x, the function ƒ(x + iy) becomes = P + iQ, 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.
It is important to remark that the demonstration does not presuppose the existence of any root; the contour may be the infinity of the plane (such infinity regarded as a contour, or closed curve), and in this case it can be shown (and that very easily) that the difference of the numbers of changes of sign is = n; that is, there are within the infinite contour, or (what is the same thing) there are in all n roots; thus Cauchy’s theorem contains really the proof of the fundamental theorem that a numerical equation of the nth order (not only has a numerical root, but) has precisely n roots. It would appear that this proof of the fundamental theorem in its most complete form is in principle identical with the last proof of K. F. Gauss (1849) of the theorem, in the form—A numerical equation of the nth order has always a root.[1]
But in the case of a finite contour, the actual determination of the difference which gives the number of real roots can be effected only in the case of a rectangular contour, by applying to each of its sides separately a method such as that of Sturm’s theorem; and thus the actual determination ultimately depends on a method such as that of Sturm’s theorem.
Very little has been done in regard to the calculation of the imaginary roots of an equation by approximation; and the question is not here considered.
10. A class of numerical equations which needs to be considered is that of the binomial equations xn − a = 0 (a = α + βi, a complex number).
The foregoing conclusions apply, viz. there are always n roots, which, it may be shown, are all unequal. And these can be found numerically by the extraction of the square root, and of an nth root, of real numbers, and by the aid of a table of natural sines and cosines.[2] For writing
| α + βi = √(α2 + β2) { | α | + | β | i }, |
| √(α2 + β2) | √(α2 + β2) |
there is always a real angle λ (positive and less than 2π), such that its cosine and sine are = α / √(α2 + β2) and β / √(α2 + β2) respectively; that is, writing for shortness √(α2 + β2) = ρ, we have α + βi = ρ (cos λ + i sin λ), or the equation is xn = ρ (cos λ + i sin λ); hence observing that (cos λ/n + i sin λ/n )n = cos λ + i sin λ, a value of x is = n√ρ (cos λ/n + i sin λ/n). The formula really gives all the roots, for instead of λ we may write λ + 2sπ, s a positive or negative integer, and then we have
| x = n√ρ ( cos | λ + 2sπ | + i sin | λ + 2sπ | ), |
| n | n |
which has the n values obtained by giving to s the values 0, 1, 2 ... n − 1 in succession; the roots are, it is clear, represented by points lying at equal intervals on a circle. But it is more convenient to proceed somewhat differently; taking one of the roots to be θ, so that θn = a, then assuming x = θy, the equation becomes yn − 1 = 0, which equation, like the original equation, has precisely n roots (one of them being of course = 1). And the original equation xn − a = 0 is thus reduced to the more simple equation xn − 1 = 0; and although the theory of this equation is included in the preceding one, yet it is proper to state it separately.
The equation xn − 1 = 0 has its several roots expressed in the form 1, ω, ω2, ... ωn−1, where ω may be taken = cos 2π/n + i sin 2π/n; in fact, ω having this value, any integer power ωk is = cos 2πk/n + i sin 2πk/n, and we thence have (ωk)n = cos 2πk + i sin 2πk, = 1, that is, ωk is a root of the equation. The theory will be resumed further on.
By what precedes, we are led to the notion (a numerical) of the radical a1/n regarded as an n-valued function; any one of these being denoted by n√a, then the series of values is n√a, ωn√a, ... ωn−1 n√a; or we may, if we please, use n√a instead of a1/n as a symbol to denote the n-valued function.
As the coefficients of an algebraical equation may be numerical, all which follows in regard to algebraical equations is (with, it may be, some few modifications) applicable to numerical equations; and hence, concluding for the present this subject, it will be convenient to pass on to algebraical equations.
11. The equation is
and we here assume the existence of roots, viz. we assume that there are n quantities a, b, c ... (in general all of them different, but which in particular cases may become equal in sets in any manner), such that
or looking at the question in a different point of view, and starting with the roots a, b, c ... as given, we express the product of the n factors x − a, x − b, ... in the foregoing form, and thus arrive at an equation of the order n having the n roots a, b, c.... In either case we have
i.e. regarding the coefficients p1, p2 ... pn as given, then we assume the existence of roots a, b, c, ... such that p1 = Σa, &c.; or, regarding the roots as given, then we write p1, p2, &c., to denote the functions Σa, Σab, &c.
As already explained, the epithet algebraical is not used in opposition to numerical; an algebraical equation is merely an equation wherein the coefficients are not restricted to denote, or are not explicitly considered as denoting, numbers. That the abstraction is legitimate, appears by the simplest example; in saying that the equation x2 − px + q = 0 has a root x = 12 {p + √(p2 − 4q) }, we mean that writing this value for x the equation becomes an identity, [12 {p + √(p2 − 4q) }]2 − p[12 {p + √(p2 − 4q) }] + q = 0; and the verification of this identity in nowise depends upon p and q meaning numbers. But if it be asked what there is beyond numerical equations included in the term algebraical equation, or, again, what is the full extent of the meaning attributed to the term—the latter question at any
- ↑ The earlier demonstrations by Euler, Lagrange, &c, relate to the case of a numerical equation with real coefficients; and they consist in showing that such equation has always a real quadratic divisor, furnishing two roots, which are either real or else conjugate imaginaries α + βi (see Lagrange’s Équations numériques).
- ↑ The square root of α + βi can be determined by the extraction of square roots of positive real numbers, without the trigonometrical tables.
