24 Proceedings of the Royal Irish Academy. 



the unit a to ca. Thus we may multiply both sides of the equation 

 (zi + So) a = so, or of any equation, by a complex number ± a ± ib, for 

 we may replace a by «<i and change if necessary a to j3, and also replace 

 a by ba, and change a to a or /3', then add the results, thus getting from 

 (Zi + s») a = sa to (.?! + .?•.) (+ rt ± ib) a = z(± a ± ib) a. 



Thus if (sj + Sj) = 2a, «i + s 2 produces the same effect as s when we 

 use it as an operator on any complex quantity. In this sense z x + So = z. 

 In the same sense if z^a = z'a, z&« = z, for, as we have just shown, 

 we may multiply both sides by z-j, getting SiZ 2 z 3 a = s': 3 o, thus SiSo produces 

 the same effect as z' when used as an operator on any complex quantity, 

 and w r e note that to get z' from Sjs, we combine the symbols in the same 

 way as if they were operating on a. 



"What we mean by (x + iy) o = is that x = 0, y = 0, for the unit 

 ia is different in kind from a. 



If (x + iy) (x' + iy') a = 0. xx' - yy' = 0, xy' + x'y = 0, so that if x' 

 and y' are not both equal to 0, multiplying the first by x' and the second by 

 y' and adding we get x (a/ 2 + y'") = 0, .'. x = 0, similarly multiplying 

 the first by - y ', and the second by o/ and adding y (x' s + y'-) = 0, .•. y = 0. 

 It follows that SjSoZj&c.a = when and only when one of the numbers 

 r,, So, s 3 , &e. = 0. 



Two square roots can now be found for any number real or complex, say 

 of x 1 ~ iy'. 



{x + iy)- = x" + iy', x- - y- = x', 2xy = y', .: (x- + y-)° = af~ + y'-, 

 taking the positive root 



*' + <f = -/•" + v'\ ••• * = ± ] ■*' + ■S*'* + v'\ y = 



I \ 9 " 9 n /-^r t. 



■L 

 2x' 



The two imaginary cube roots of unity have from this point of view 

 precise definite meanings, as either specifies an operation which when 

 performed three times on any complex quantity reproduces the quantity 

 itself. 



Fractions. — What we mean by the fraction — ■ — ~, is a complex 



x' + uj r 



number such that when we multiply it by x 1 + iy', we get x + iy. 

 Assuming it to exist, we call it s, then 



x + iy 



x + %if 



