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 x^n - a = 0 (a = [alpha] + [beta]i, 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 / [alpha] [beta] \ [alpha] + [beta]i = [root]([alpha]^2 + [beta]^2) ( ---------------------------- + ----------------------------i ), \[root]([alpha]^2 + [beta]^2) [root]([alpha]^2 + [beta]^2) / there is always a real angle [lambda] (positive and less than 2[pi]), such that its cosine and sine are = [alpha] / [root]([alpha]^2 + [beta]^2) and [beta] / [root]([alpha]^2 + [beta]^2) respectively; that is, writing for shortness [root]([alpha]^2 + [beta]^2) = [rho], we have [alpha] + [beta]i = [rho](cos[lambda] + i sin[lambda]), or the equation is x^n = [rho](cos[lambda] + i sin [lambda]); hence observing that (cos [lambda]/n + i sin [lambda]/n )^n = cos[lambda] + i sin[lambda], a value of x is = [root n][rho] (cos [lambda]/n + i sin [lambda]/n). The formula really gives all the roots, for instead of [lambda] we may write [lambda] + 2s[pi], s a positive or negative integer, and then we have