Introduction. 5 



9. Incommensurable Numbers. Algebraic numbers* may 

 be divided into two kinds, depending upon the relation which 

 they bear to the unit or unity. If a number has a common 

 measure with unity, it is called a commensurable number. Thus 

 7 is a commensurable number ; also f is a commensurable number, 

 since one quarter of the unit is a common measure between f and 

 unity. Commensurable numbers thus include both integers and 

 fractions. If a number has no common measure with unity, it is 

 called an incommensurable number. Thus v^ 2 is incommen- 

 surable. A little consideration will show that v^ 2 cannot be an 

 integer nor a fraction. It is not an integer because (0)^=0, 

 (1)^=1, and (2)^=4, and there are no integers intermediate be- 

 tween these. It cannot be a fraction, for if possible suppose that 



some irreducible fraction, represented by-^, equals v^ 2 . Then 



^ 2 — -r-, or squaring, 2=-^, which is absurd, for an integer 



cannot equal an irreducible fraction. Therefore >/ 2 is not a 

 fraction. But it is an exact quantity, for we can draw a geomet- 

 rical representation of it. Take each of the two sides, CA and 

 CB, of a right angled triangle equal to i. Then AB, the hypoth- 

 enuse, will equal v^( i )M-(i)'= ^ ~2 Thus A^ 

 v^ 2 is the exact distance from A to B, 

 which is a perfectly definite quantity. 

 Thus the idea that incommensurables are 

 indefinite or inexact must be avoided, (l) 

 This notion has arisen because the frac- 

 tions we often use in place of incommen- 

 surables, such as 1. 4 1 42+ for >/ 2, are 

 merely approximatioyis to the true value. (^ (x) ~J^ 



We now give a property of incommensurable numbers which 

 will serve to make their separation from the class of commensur- 

 able numbers (integers and fractions) more apparent. It is that 

 an incommensurable yiumber whe^i expressed in the decimal scale 

 never repeats, while a commensurable number so expressed always 

 repeats. 



* As here used the term Algebraic number does not include the so-called imaginaries, 

 which, strictly speaking, are not numbers at all. Imaginaries ai-e treated in Chapter 1, 

 Part 11. 



