8 Algebra. 



17. In Algebra quantities are represented by letters, but a letter 

 is just as apt to represent a quantity to the left of o in the above 

 scale as it is to represent one to the right of o ; so that, while in 

 the case of a numerical quantity, i. e. one represented by figures, 

 we can tell whether the quantity represented is positive or nega- 

 tive by the sign preceding it, yet, it the case of a literal quantity, 

 /. e. one represented by letters, we cannot tell by the sign before 

 it whether the quantity represented is positive or negative. 



If we speak of the quantity 5 we know that it is positive, but if 

 we speak of the quantity a we do 7iot know by the sign before it 

 whether it is positive or negative. 



We know that —5 is negative, but we do 7iot know that —a is 

 negative. 



A mifius sign before a letter always represents a qna?itity of the 

 opposite kind from that represented by the same quantity with a plus 

 sign or no sig7i at all before it. Thus, if «=3, then — <2=— 3, and 

 if a=— 3, then —a—T). 



18. Looking at the above scale it is evident that of any two 

 positive quantities the one at the right is greater than the other or 

 the one at the left is less than the other, e. g. io>6 or 6 < 10. 



Now it is found convenient to extend the meaning of the words 

 * ' less than ' ' and ' ' greater than ' ' so that this same thing shall be 

 true throughout the whole scale. 



Thus we would say that 



— 5< — 3 and— 2<o. 

 It should be carefully noticed that this is a technical use of the 

 words ' ' greater than ' ' and ' ' less than ' ' and conforms to the pop- 

 ular use of these words only when the quantities are positive. 



Of course it would be wrong to say that — 2 is less than o if we 

 use ' ' less than ' ' in the popular sense, because no quantity can be 

 less than nothing at all, in the popular sense of " less than." 



In objecting to the use of the words " less than " in the popular 

 sense. Prof. De Morgan, one of the great mathematicians of Eng- 

 land, says : " The student should reject the definition still some- 

 times given of a negative quantity that it is less than nothing. It 

 is astonishing that the human intellect should ever have tolerated 

 such an absurdity as the idea of a quantity less than nothing ; 



