26 Proceedings of the Royal Irish Academy. 



and the answer to the question is = — *" ~ u->, or I pay for the dozen 



/->Q- 



eggs hy receiving one halfpenny, and either giving or receiving v pounds 



of corn. 



One fundamental group of -i units referred to at the beginning of 

 the treatment of complex numbers is a set aa'fifi', in which a/3 are units of 

 length measured along a line OA, and a'i$ are units of length measured 

 parallel to a line OB meeting OA at an angle Q. A complex quantity 

 (x -r iy) a for this group may be represented by measuring 031 = .ra from 

 along OA and MP from M = yd parallel to OB. The lengths xa, yd 

 are what are called the coordinates of P referred to the oblique axes OA, 

 OB. Any complex number x -f iy provides us with the coordinates of a 

 point P referred to given oblique axes, and conversely the position of P gives 

 us by its coordinates x, y a complex number x + iy. In this sense P is a 

 geometrical representation of the complex number. To add two complex 

 numbers represented by P and Q, we draw from P a line equal and parallel 

 to and in the same direction as 00. 



Any complex number x + iy may be written in the form 



r (cos 6 + i sin 0), 



where ;• is taken to be a real positive number, is called the modulus of 

 - iy, and is denoted by \x ~ iy\. The angle is called the argument or 

 amplitude of x + iy, and has an infinite number of values, as we may 

 take - '21;t instead of without altering .r - iy, where £ is an integer. 

 "When the axes OA, OB are taken to be at right angles, the angular 

 coordinates r, 6 of a point P, namely the length OP taken to be positive, 

 and the angle POA, also represent the modulus and argument of the complex 

 number represented by the point P. "Writing two complex numbers s, z in 



the form 



r (cos - i sin 0), r* < cos 6' - i sin 6 ' r 

 their product 



ss = r/;cos(0 + 0') + i sin (0- ■ 



by de Moivre's theorem. Thus the modulus of the product of z, z' is the 

 product of their moduli, and the argument of the product is the sum of 

 their arguments, to which we may add, however, ± 2kn. As 



— . . 



cos ^ - i sin 





when OA, OB are at right angles, the point P' which represents i{x - iy) 

 may be obtained from the point P, which represents x + iy by rotating 



