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.[3]

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.[4] For writing

α + βi = √(α² + β²) { α+ βi },
√(α² + β²) √(α² + β²)