FEOM NUMBER TO QUATERNION. 73 



15 



7 and by for instance, come respectively to multiplications by 



1 , 101 

 - and — — 

 7 15' 



Directive Quantities — Complex Quantities — Algebra. 



So far we have only been dealing with arithmetic, for the study 



of absolute magnitudes comes to the study of their tensors, that is 



to say of numbers, which is precisely the aim of arithmetic. 



But the absolute magnitudes are not the only ones to be con- 

 sidered in science, other magnitudes such as time, temperature 

 etc., which can vary in two opposite directions, have also to be 

 considered. Such magnitudes can be symbolised by straight lines 

 or by translations as the preceding ones, but by straight lines 

 capable of two directions. 



Thus the notation AB will represent as before the translation 

 of a movable point from A to B, but BA will represent a trans- 

 lation from B to A in the opposite direction to the first one. 



The direction will be determined without ambiguity if we know 

 the angle made by AB or BA with the standard direction from 

 left to right in taking always our lines horizontal. 



AB (Fig. 3) makes an angle zero or 



^ jsyuadranbs more generally 4 k quadrants (k being 



/ \ a whole number) measured by 2 k in 



but BA makes an angle equal to 2 



quadrants (measured by tt) or more 



generally makes an angle equal to (4 k + 2) quadrants measured 



by (2& + 1)tt. 



If tensor AB = a we can represent AB by a or simply a f and 

 BA by a 7Cm 



That angle or its measure is called the argument* of the 

 quantity considered. 



* Another word is also wanted instead of argument as it is used for 

 some other purposes, but the words norm and amplitude sometimes used 

 respectively instead of square of modulus and argument are subject to 

 the same objection. 



