6 Algebra,. ^^ ... .^^^^. ; 



Thus, 75=75.0000000000+ repeati7ig the o. 



|-= .5000000000+ repeating ihe o. 

 i= -3333333333+ repcati7ig the 3. 

 yYT= .279279279279+ repeating the 279. 

 \/ 3 = 2.7320508+ never repeating , 

 "^20= 2. 71 44 177+ 7tever repeating. 

 -= T). 1/^1 ^^26+ never repeating. 

 The student should endeavor to get a fair notion of what is 

 meant by an incommensurable number. It is a difficult idea to 

 grasp at once, but it is one which the student should continue to 

 consider until the conception takes a definite and rational shape. 



POSITIVE AND NEGATIVE QUANTITIES. 



10. In Algebra we are often called upon to distinguish between 

 quantities which are directly opposite each other ; as, for instance, 

 degrees above zero from degrees below zero on a thermometer scale,, 

 distance north of the Equator from distance south of the Equator, 

 distance east of a given point from distance west of the same given 

 point, etc. 



The distinction is made by means of the signs + and — , e. g.^ 

 if +10° means a temperature of 10° above zero, then —10° would 

 mean a temperature of 10° beloiv zero, and if + 10 miles means 10 

 miles «^r/// of the Equator, then — 10 miles would mean 10 miles 

 south of the Equator, and if +10 rods means 10 rods east of a 

 given point, then — 10 rods would mean 10 rods west of the same 

 given point, and if +10 be ten units of any kind in a7iy sense, 

 then —10 would be ten units of the sa7ne kind in just the opposite 

 sense. 



These two kinds of quantities are called /^«VzV^ and negative. 



11. The distinction between positive and negative quantities is 

 made by means of the same signs as are used to denote the opera- 

 tions of addition and subtraction, and it might seem that it is un- 

 fortunate and unnatural that the same signs are used in these two 

 ways. It may be unfortunate, but it is not unnatural, as w^e pro- 

 ceed to show. 



12. Suppose that, by one transaction, a man gained $500, and 

 by another he lost $700 ; then he lost all he gained and $200 more, 



