Common multiplication makes it obvious that {cos x + sin x V( -1)} {cox y + sin y V( -1)} =cos (x+y)+ sin (x +y) V(-1) for all real values of .e and y; so that if we represent coo + Rio x. -1 ) by nx we have 7,X x Now in Iltsomtst. Tnr.outy It is proved that this equation cannot be universally true without giving as a consequence (nx)" n(ex), for all values of to, whole or fractional, positive or negative. We have then ices x+ sin x V(-I)}'=cos ex+ ain tax . ( —1) .... (3) an equation which goes by the name of De Muivre's Theorem. It is the key of the present subject.
Let it now be required to raise the nth power of a + b si( -I), being integer or fractional, positive or negative : this includes every rase of raising a power, extracting a root, performing both operations, and taking the reciprocal of any result. Reduce a+ b ../(-1) to its equivalent form rn(0+ 21w), or r {cos (0+ 2kr) + sin (0+21r) . V(-1)}, whence is ire(0 + 24sr)}' or en(a0 + 2akte), or in+ Icoa (me + 2nkr)+ (a0+2mkr). V( - DI,in which is found by purely arithmetical operation, and cos (e0 + 2nisr) and sin (e0+ by aid of the trigonometrical tables. So many Extinct values as the variation of k enables us to give to eit +2nkr, so many values do we find of a +bs/(-1)}'. Two angles are distinct when they are unequal, and do not differ by 2w or a multiple of 2w.
Firstly, let ta be a whole number, positive or negative, then 2-ak is always an integer oven number, and there is only one value, namely, :cos e8+sin s0. V(-1)}.
ext, let n be a fraction in its lowest terms, and, choosing an example, 4 sly a = T. Let us examine all the values of from k.• -5 to U.- +5, making Ak = WO+ 2nkr.
4 4 32 4 24 8w, - 5 A.., 4 16 4 8 4 4 8 4 16 4 24 4 32 4y= 39+ L Tir, A. TM, 8w.
Here it would seem as if from this stet of the possible values of we get eleven distinct values of the fifth root of the fourth power of + b V (- 1). But a moment's inspection shows that a_s, A,, are lot distinct in effect, since they differ by multiples of 2r; neither are s_s and s„ nor and A., nor a_s and nor A...1 and a,.
Also it will be found that for every value of k • Ak Ak + IR, &c., are all 'angles which differ, each from its predecessor, by 2r; so that there are but five distinct angles in the whole series, which may be found by taking Ak, Ak+1, Ak+ri, Ak +3, and with any value of k positive or negative. And generally, if n be a fraction whose deno
minator (when the fraction is reduced to its lowest terms) is q, it will be found that there are q distinct values of la + b si(-1)}" and no more.
The most important cases are those in which r=.' 1, or 0+ M=1, in which cos sin 0 V(--1) may represent the expression. And of this particular case, the most important more particular cases are 0=0 cos 0+ sin 0.../(-1)=1 0=w cos (INA—I)... —1 0= it cos 0+ sin 0V(-1)= V(-1) Ossir cos 0+ sin — (-1)Of these again, the two first are the most important.
Let st : q, and let the question be to find the q qth roots of 1. Putting unity in the form cos 2kw + sin 21w . sl( - 1), all 21sr 2k r these roots are the distinct values of cos - +sin - V( -1) 2r 2ror { 7 cos - cos -- + sin - V(-1) .
r 9 9 2r 2 9 w + sin . —1)— a, cos - -sin - . V(-1)=0.
7 Then at3=1, as will be found by multiplication, and : = since al =1. Consequently, since the series of powers of a, positive and negative, are successions of qth roots of 1, the series of powers of /3 will be the same ; and we may therefore select these roots at convenience from either series, or partly from one and partly from the other. Thus, if we would have the ten tenth roots of unity we may form them in pairs, as follows : 2. Or 2 . Or ce and $ give ± sin V(-1) both = I 2r 2w and or . . cod To ± sin 4w 4w and or . . cos ± sin Cr 6v and or . . cos ± sin Th. s/(-I) 8w 8wa' and or . . ± 'in A-3) 10r 1 Oral and or . . cos-To- ± both = - 1 Of, these twelve only are distinct, giving the ten tenth roots required. In this way the following theorems may be easily demonstrated.
1. The (2nt)th roots of unity are + 1, -1, and the 2»t -2 quantities contained in 2k w sin V('-') • for all values of k, from k=1 to k=ni-1, both inclusive.
2. The (2m+ 1)th roots of unity are 1 and 2m quantities con. tattled in 2kr 21:cos 2m+1 ± sin V(-1) for all values of from k=.1 to k=m, both inclusive.
3. If u be one of the qth roots of unity, are also qtli roots, but do not contain all the I roots, unless n be made from a value of k which is prime to q. Thus, if q=12, and k=1, we get 2w 2w a=cos + sin V(-1)the list of roots is complete in 3, a, .... and 15 1, is a, itc.