8 NUMBEK. [INT. I. 



and /3. Whatever the values of a and /3 we may always find a 

 corresponding point, and to every point in the plane there corre- 

 sponds a single complex number, including the real and pure 

 imaginary numbers as particular cases. As each of the real 

 numbers a and /3 may independently assume the value of any of 

 the single infinity of real numbers, there is said to be a double 

 infinity of complex numbers, or a complex number has two degrees 

 of freedom. The distance of the representative point from the 

 origin is called the modulus of the complex quantity and denoted 

 by | a | = + Va 2 + ft* since it includes as a particular case the 

 absolute value of a real number. The angle that the radius vector 

 from the origin makes with the X-axis is called the argument of 

 the number. This representation of complex numbers in the plane 

 was proposed by Argand and Gauss*. 



The threefold complex quantity a cd + {3j + <yk, not being 

 capable of representation in a plane, may in a similar manner be 

 represented in space. If we take three mutually perpendicular 

 axes, points at equal distances from their intersection will repre- 

 sent the three units i, j, k. Multiples of these by real numbers 

 will be represented by points on the axes of X, Y and Z, and any 

 complex number ai + ftj + yk may be represented by a point 

 having the rectangular a?, y and z coordinates a, /3, 7. For every 

 complex number we may find a point, and to every point there 

 corresponds a complex number. As each of the real coefficients 

 a, $, ry may independently assume any of a single infinity of values, 

 the complex number has three degrees of freedom, or there is a 

 triple infinity of such complex numbers. The distance of the 

 representative point from the origin was called by Hamilton the 

 tensor of the complex number. We may apply the term modulus 

 to the tensor, and use the symbol 



In this book the arrangement of the axes of X, Y, Z will 

 always be such that the motion of a right-handed screw along the 

 axis of X will turn the Y axis toward the Z axis. This will be 

 called right-handed cyclic order, Fig. 1. 



* Argand, Essai sur une maniere de representer Us quantites imaginaires dans 

 les constructions geometriques, Paris, 1806. 



Gauss, " Theoria residuorum biquadraticorum, commentatio secunda." Werke, 

 Bd. ii., p. 169. 



