EXPONENTIAL EQUATIONS.] M ATHEM ATICS. TRIGONOMET R Y. 



631 



If we subtract this from the result last found we shall 



obtain . and .'. multiplying that difference by 4, we 



4 

 shall obtain the value of tr. required. Thus, 



789582239404 



004184076002 



785398163402 



4 



3-141592653608 



The calculation is carried to 12 places of decimals, but 

 is not trustworthy beyond the first 10. 

 Hence (to 10 places of decimals) 



TT = 3-1415926536 



If we had wanted to obtain the value of a- to a larger 

 number of places of decimals, we should have had to 

 carry the calculations throughout to a correspondingly 

 greater extent. 



Thus, we can show that 



v = 3-141592653589794* 



We can very conveniently apply De Moivre's formula 

 to the solution of Binomial Equations, i.e., equations of 

 the form x"+ a = 



(54). To explain what we mean by the roott of a Binomial 

 Equation. 



If we take the case x' + a = 0, and if a be 6", then 

 the equation becomes 0" + 6" = 0, and if x = by. x" = 

 6* y" , and then the equation becomes 



r + i = o 



where the 1 may be either positive or negative. It 

 might seem at first sight that this has only one root, viz., 

 unity, but to conclude BO would be an error, as a little 

 consideration will make quite plain ; for take the case 



x* 1 = 



1, or 1 ; i.e., has two values. And 

 x 3 1 = 



Then x is either 

 if we take the case 



Then because 



we shall have 



(x 1) (t* + x + 1) = 

 whence x will have values corresponding to 



x 1 = 

 x 2 + x + 1 = 



From the former of them we find that x = 1, 

 and from the second that 



x* + x + i = -? 



1 / 3 

 /.x = 4 



And hence x has the three values. 



1. 



1 



2 2 



which are three roots of the binomial equation 



x 3 1 = 

 And hence we manifestly have that the roots of 



' a 3 = 



are 



, 1 V 3 



In like manner 



X " + 1 . 



lias n different roots, which indeed follows from the 

 general principle, that every equation of the n" 1 degree 

 lias n different roots, real or imaginary. 



In connection with this, the rtndent may advantageously pemw the 

 remarks of Professor Younit, in Chapter IV. on Geometry, in which he 

 referi to Uw quadrature of the circle. (Soe p. 577). 



(55). Tofmd the roots of the Equation, x n + 1 = 0. 



Since, cos. (2 p + 1) TT + V~~^\ sin. 2 p + !)TT= 1 

 whatever be the value of p. provided it bo an integer. 

 And since 



x"= 1 

 we must have 



x" = cos. (2p + 1) TT + V 1 sin. (2p + 1) TT. 

 .". x=> (cos. 2p 



Hence cos. -**jj . IT + -/ 1 sin. jr. is a 



root of the binomial equation whatever integral value we 

 may give to p. 



We shall prove that there are n different values of this 

 formula, corresponding to the different values dip, viz. : 

 0, 1, 2, 3. ... n 1, and that there are no more than n 

 values. 



(a). Let p and q be two values, each < n ; and if pos- 

 sible let 



Ileuce equating possible and impossible terms, 



2/>-f-l 2o + l 



cos. . = cos. - IT, 



n 



sin. 



n 



TT = sin. 



Hence the angles must differ by some multiple of 2 w, 

 say by fc X 2 a-. 



. . 



TT =3 - yr -t- J K Tr. 

 n n 



.'. p 2 = /;.H. a multiple of n, 



and therefore either p or q must be greater than n. 

 Hence all the values of x which correspond to the values 

 otp^n, viz., 0, 1, 2, 3, .... n 1, are different from 

 each otht-r. 



(6). Again, if we give p a value r greater than n, then 

 the corresponding value of x will be the same as one of 

 those which is given by some value of p, that is less 

 than n, as for instance q. 



For suppose p = kn + q- 

 where k is a whole number, and q is less than n. 



2 o 

 Then, cos. 



Hence by giving to p. successively the values of 

 0.1.2.3 ..... n 1. in the formula 



. 



we obtain all the roots of the Equation 

 x + 1 = 0. 



Thus. To find the roots o the Equation 

 x + 1 = 0. 



the roots are given by the formula 



and are therefore respectively 



