ALGEBRA 



Ques. How far apart are they? Ans. 24 

 points. 



Ques. What direction does John go? Ans. 

 Negat 



Ques. How far? Ans. 24. 



Then answer showing distance and direction. 

 What is the difference between John's and 

 Mary's financial standing? Ans. 24. 



Between Mary's and John's? Ans. +24. 



The first means that John must lose or get 

 rid of in some way, 24. The second means 

 that Mary must gain 24. 



These would appear when set down in 

 ordinary subtraction 



(1) 8 (2) +16 



+16 - 8 



^24 +24 



Show difference between 7 and +20. 

 -Z7 



-15 



188 ALGEBRA 



(1) Go from +20 to 7. 



(2) Go from 7 to +20. 

 The thermom- 

 eter was at 85 



at noon, and at 

 70 at 6 P.M. 

 Change? 15. 



Thermo m- 

 eter was at 10 

 below at mid- 

 night and at 30 

 above at 10 A. M. 

 What was the 

 change? 



There was a 

 rise of 40 : 



Ans. 27. 

 Ans. +27. 



+30 

 10 



+40 



-7 



4-20 



+27 



Such problems 

 may have infinite 

 variety. A.H. 



+80 



+70 



+30 







-10 



In this treatment of the subject it is possible 

 merely to explain fundamental principles, and 

 to show how simple and reasonable the boy 

 or girl can find this hitherto unknown science. 

 There are many new things to be learned that 

 were not treated in arithmetic; the reason for 

 the existence of every new principle is not at 

 all difficult to understand, and if the young 

 student masters each principle in turn the 

 entire subject may become a delightful recre- 

 ation. 



Signs and Symbols. The signs used in arith- 

 metic are carried into algebra without change 

 of form and with meaning changed only in one 

 particular : 



+ (plus) indicates addition; 



(minus) indicates subtraction, and it has 

 also a new significance, for it designates nega- 

 tive number; 



X (times) indicates multiplication; 



-j- (divided by) indicates division, and 



= (equals) is the sign of equality. Whatever 

 appears on one side of this sign in an algebraic 

 problem is exactly equal in quantity, number 

 or amount to that which appears on the other 

 side of it. See Simple Equations, below. 



In algebra, parentheses, braces and brackets 

 are called signs of aggregation, because every- 

 thing within a pair of any of them is to be 

 treated as a single quantity, which is to be sim- 



The Foundations of Algebra 



plified (reduced to its simplest expression) be- 

 fore being incorporated into other parts of a 

 problem. Their treatment may thus be ex- 

 plained : 



[12+{4+5 (5 3)+4} 4]=what number? 



We must first simplify the inside group (5 3) ; 

 after doing so the problem is stated in new 

 form: 



[12+ {4+5 2+4} 4] = ? 

 Again simplifying the term within the inside 

 signs, the problem becomes: 



[12+114] = ? Ans. 19. 



This problem is purely arithmetical. When 

 applied to algebra there is no change in prin- 

 ciple. Having observed the solution above, 

 solve the following, which is purely algebraic: 



[5a+6a+ {5a a+ (3a+4a) } a] = ? 

 If a=4, what is the numerical value of the 

 series? 



Coefficient. The beginner in algebra at once 

 finds a much-used term not employed in arith- 

 ment the word coefficient. As usually under- 

 stood it means any number or letter placed be- 

 fore another letter, and it indicates multiplica- 

 tion; a coefficient, then, is a multiplier. Thus, 

 in the term 5a, 5 is the coefficient of a, and 

 indicates that the value of a is to be taken 5 

 times. After becoming a little more familiar 



