177 



ROOT. 



ROOT. 



178 



Let us now take a quantity of the form a + >>\ ( 1). Assume 

 r cos =, r sin t = b, which gives 



b^ 



' ' 



Let us choose for r, which is called the mntlalni of the expression, 

 the positive value \'(lS + <i'). We can then always make the angle 

 give the equation 



(-l) = r cog + r sin 9 V( 



(1) 



identically tnie. If a and A be both positive, must lie between 

 and a right angle, or between and \v [AxoLE] : if o be positive and 

 t negative, must lie between j and 2w : U 6 be positive and a 

 negative, must lie between $w and : and if bth be negative, 8 must 

 lie between * and }-. Thus reducing angles to degrees and minutes, 



2 + 3V(-l)= V13{cos 56* 19* + sin 56 19'. v (-l>} 

 l) = VlScos 123 41' + sin 123' 41'V(-1)} 



2-3 v (-l) = 

 _2-8V(-l) = vl3{cog236M9'+siii236' > 



Generally, if a and 6 be positive, and if, returning to the arcual mode 

 of meaxuring angles, be that angle which lies between and IT and 

 ha* It: a for its tangent, we must use i for a + 6v( 1), * - t for 

 -a + 4V(-l>, 2-fora-AV(-l), nd + for -a A\/(-l). 



Again, since * (- 'lie* has the same sine and cosine as I, when t is any 

 whole number, positive or negative, if we take f so an to satisfy (1), 

 we find that the following is also satisfied : 



fur all integer values of I positive or negative, but for no frac- 

 tional value of t whatsoever. This and various other result* of 

 common trigonometry should be familiar to every student who 

 attempts the present subject. 



Common multiplication makes it obvious that 



{cos-z + sin x. V(-l)} { cos y + sin y vX-ljJooi (* + #) + 



ain(*-r,)V(-l) 



for all real values of x and y; so that if wu represent cos x + 

 un x . v( 1' by IK we have ijjc xijy = TI(X + y ). Now in BINOMIAL 

 THKOBKM it is proved that this equation cannot be universrUly true 

 without giving as a consequence (IK)' 1( <), tor all values of it, 

 whole or fractional, positive or negative. We have then 



{cos .r + iin jr V( !)}"" 



. v ( - 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 +6 */( !), 

 being integer or fractional, positive or negative : this includes every 

 case of raising a power, extracting a mot, performing both operations, 

 and taking the reciprocal of any result. Reduce o + 6v(l) to its 

 equivalent form nf,t -t- 2for), or 



r {cos < + 2i) + sin (t+tkw) . V(-l)}, 

 whence { + 6 V( -1)}' is {r*l+U-w)}' or I-IT> + 2t), or 



-r"{c 



in which r* U found by purely arithmetical operation, and cos (( 2whr) 

 and sin ( 4- 2*Jnr) by aid nf the trigonometrical tables. So many 

 distinct values as the variation of k enables us to give to + -tut*, so 

 many values do we find of {a + 6V(- 1)J". Two angles are distinct 

 when they are unequal, and do not differ by 2 or a nitilti|>l>- of -i*. 



Firstly, let be a whole number, positive or negative, then 2ml- is 

 always an integer even number, and there is only one value, namely, 



Next, let be a fraction in its lowest terms, and, choosing an example, 

 ay - y Let us examine all the values of k, from i S to 

 It" -r 6, making A* 



32 



* 1 48 4 



*-= 6*-T*' *- = 6*- 5*' *-6*> 



24 



4 1 4 24 4 32 4 



** = 5 * + T ' * " 6 ' + T ' * = C * * T *' A = 5 ' + 8 - 



Here it would seem aa if from this set of the possible values of k, 

 we get eleven distinct values of the fifth root of the fourth power of 

 > Ay ( 1). But a moment's inspection shows that A_4,A a , A,, are 

 not ilixtinct in effect, since they differ by multiples of 1* ; neither 



AMS A*D C1. Div. vol.. vn. 



are A_I aud A,, nor i_ 3 and A,, nor A_ 2 and A 3 , nor A_I and A.. 

 Also it will be found that for every value of k 



A *+.S Ai+'O, A t tl.i, &C., 



are all -angles which differ, each from its predecessor, by 2ir ; so that 

 there are but five distinct angles in the whole series, which may be 

 found by taking A, A t+ i, A 1+5 , A* 4.3, and A i+4 , with any value of 

 1- 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 {o + 6\/( l)J"and no 



more. 



The most important cases are those in which r = l, or a 2 + J=l, in 

 which cos + sin ^( 1) may represent the expression. Ami of thia 

 particular case, the most important more particular cases are 



6=0 



8=4 cos + sin *\/( 1)= V( 1) 



Of these again, the two first are the most important. 



Let = 1 : q, and let the question be to find the q qih roots 

 of 1. Putting unity in the form cos 2r + sin 2Xir . V( 1), all 



2iir 2k f 

 these roots are the distinct values of cos +sin V(-l) 



f 2 2 i * 



or < cos +sin v( 1) t 

 I 9 9 J 



Letcoe + sin . V( 1 ) = cos - sin . \/( !) = # 

 Then o = l, as will be found by multiplication, and o* =(3~* : 

 *=o*i = ^-*-, since a'=l. Consequently, since the series of 

 powers of a, positive and negative, are successions of qth roots of 1, 

 the series of powers of ft 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 tent h 

 roots of unity we may form them in pairs, aa follows : 



2 . OT 2 . Or 



o and ff> give cos -jjp + sin -jjj- V( 1) both = l 



2r 2r 



o 1 and ' or , . . cos j^ + bin yg V( 1) 



o^andS 5 or a* . . cos r -f- sin -^. V( 1) 



o 3 and /9* or a? 



a 4 and /3< or of 



65 



10 



15 



8m - ^~ 



6 

 10' 

 Sir 

 10' 



a* and fl> or * . .cos -JQ- + sin -^ V(- 1) both = 1 



Of. these twelve forms, ten only are distinct, giving the ten tenth 

 root* required. In thia way the following theorems may be easily 

 demonstrated. 



1. The (2*t)th roots of unity are +],-!, and the 2; 2 quantities 

 contained in 



'2kr 2ir 



for all values of t, from i= 1 to i- = w ], both inclusive. 



i. The (2m+l)th roots of unity are 1 aud 2m quantities con- 

 tained in 



Zkr 2k* 



008 8 



for all values of k, from i = I t<> i = m, both inclusive. 



3. If n be one of the '/tli roots of unity, /i 3 , ^', .... are also </tli 

 roots, but do not contain all the 7 roots, unless p be made from a value 

 of k which is prime to y. Thus, if j = 12, and k = I, we get 



2r 2r 



o-cos j^ + sin J2 V(-l) 



the list of roots U complete in 1, a, a?, a 9 , .... a", and a 1 -' is 1 , a" 

 is a, 4c. 



But if we make t=8, or take a 9 for /*, we have 



M 3=o'=a, /. J = 54 =l, M 4 =a"=a", M 3 = a*= < , &c , 

 no that we get no roots from this series but a 8 , a 4 , 1 , which are only 

 the three cube roots of 1 (cube roots are among twelfth root,-- 1. But 

 choose a* (5 in prime to 12) and its successive powers are a', a"', a 1 ' or 

 a*, a or o', o a or a, a 30 or a 6 , a 1 * or a", o 40 or a", a 4 * or a 9 , n 5 " or a ? , 

 a 4 * or 7, a** or 1, after which the series recurs in the same order. 



4. If m be any factor of q, all the mth roots of unity are among the 

 0th roots. Thus, if q : m=v, and if a be the first of the series of gth 

 roots, the roth roots are a* , a**, .... " or 1 . For (a' )" = a r * =' = !, 

 tc. All thoe powers of a which have exponents prime to q, may be 

 called primary <?th roots of unity : thus the primary 12th roots are 

 a, a, a', a". 



