Fry — Real and Complex Numbers as Adjectives or Operators. 25 



Multiply both sides by (Y + iy) (»' - iy'). 



.-. z (x + iy) (x' - iy') = (.r + iy) (x - iy') ; 

 •■• (* 3 + y' 2 ) z = xx + yy + {xy - xy') i ; 

 xx' + yy + (x'y - xy') i 



If we multiply by x' - iy', we verify that we have obtained the 



i i * + iy 

 complex number —. ~ 



;c + iy 



If s = -, then s3o=i 1; /. ss = £ 3 = s 1 s 3 , .-. s = — ", or a complex 



fraction may be multiplied above and below by the same complex number. 



As we found above two square roots of the form ± (x + iy) for any 

 complex number, it follows that any cpuadratic s 2 + 2az + b, where a and b 

 are complex, may be expressed as the product of two complex factors 

 (r + w) (s + w'), where a = a + yd- - b, ia = a- ,/cc - b. In particular 

 s 2 + n- = (z+ ia) (z - ia). 



To apply these complex numbers to solve problems. Taking the same 

 problem as before, but altering the word " lowers " to " increases " in order 

 to make the solution complex, we want to solve : — "What are eggs a dozen 

 if two more in a shilling's worth increases the price a penny a dozen ? 



Here with the pair of units ai(3i for eggs we associate two others, 

 say a\ an apple I am to give away, fi\ an apple I am to receive, and with 

 the pair a?, /3 2 for pence we associate two others, say a' z a pound of corn I 

 am to part with, j3' : a pound of corn I am to receive, thus getting two 

 fundamental groups each consisting of 4 units properly arranged and in 

 cyclical order aia'i/3i|3'i, a z a' 2 f3 2 j¥ 2 respectively. 



The price of an egg is a complex quantity ;a 2 , the price of a complex 

 number n of eggs is nza-., so that if n is the number of eggs I get for a 

 shilling, the problem states 



nza, = 12a,, (n + 2) (z + -^0, = 12a, : 



■'• »-t« and (T f2 )( s + r2) = 12 - : 

 •'• (= + 6 )( s+ b) = 6 - 



. - _ - 1 ± i v/287 . 

 •• *- ^ ' 



[4*] 



