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.
Algebraical Equations.
11. The equation is
xn − p1xn−1 + ... ± pn = 0,
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