LESSONS IN ALGEBRA. 



61 



EXERCISE 1. 



Give the algebraical expression* for the following 

 in words : 



1. The product of the difference of a and h into the sum of 6. e, 

 and J, is equal to 37 times m, added to the quotieut of I divided by 

 the um of 'i aud I. 



2. The sum of a and b, i to the quotient of b dirided by e, ai the 

 product of a into c, is to 12 time* k. 



3. The cum of a, b, and c, divided by six time* their product, i equal 

 to four time* their sum diminished by d. 



4. The quotient of 6 divided by the sum of a and b. is equal to 7 

 timea d, diminished by the quotient of b, divided by 96. 



34. We now give an example of the method of writing oat 

 algebraical expressions in words. 



EXAMPLE. What will the following expression become, when 

 words are substituted for the signs P 



AlXJKBRAlCAL EXPRESSION. ~ = abc - 6m + -^_. 



h + C 



STATEMENT IN WORDS. The sum of a and b divided by h, is 

 equal to the product of a, 6, and c, diminished by 6 times m, 

 and increased by the quotient of a divided by the sum of a and c. 



EXERCISE 2. 



Write out the following algebraical expressions in words : 



= (a + A) (b - c). 



ci 



3 -r 16 c) 



h + dm' 



35. At the close of an algebraic process it is often necessary 

 to restore the numbers for which letters have been substituted 

 at the beginning. In doing this the sign x must not be omitted 

 between the numbers, as it generally is between factors expressed 

 by letters. Thus if a stands for 3, and 6 for 4, the product ab 

 is not 34, but 3x4, i.e., 12. 



EXAMPLE. If a = 1, 6 = 2, c = 3, and d = 4 what is the 



numerical value of the expression \- c + ? 



b a 



Substituting the value given above for each letter, the alge- 

 braical expression + c + becomes in figures - + 3 + 



2x34 6 



j or-+3 + -or2 + 3 + 6, which is equal to 11. 



EXERCISE 3. 



Find the values of the following algebraic expressions, sup- 

 posing a = 3; 6=4; c = 2 ; d = 6; m = 8; and n =10 : 



ad 



1 + a +mn. 



b + mm be + n 

 cd + ~~3d~ 







b + ad b + 4f 



X 5 +bcw ob~' 



Vf 3t>i do 



4. bm + 



5. cbm + 



m b 



+ 2n. 



POSITIVE AND NEOATITS QUANTITIES. 



36. A POSITIVE or AFFIRMATIVE quantify is one which is to 

 be added, and has the sign + prefixed to it. (Art. 11.) 



37. A NEGATIVE quantity is one which is required to be SUB- 

 TRACTED, and has the sign prefixed to it. (Art 10 and 11.) 



When several quantities enter into a calculation, it is frequently 

 necessary that some of them should be added together, while 

 others are subtracted. 



If, for instance, the profits of trade are the subject of calcu- 

 lation, and the gain is considered positive, the loss win bs 



negative; beoauM the latter moat b subtracted from the 

 former, to determine the clear profit If the sums of a book 

 account are brought into an algebraic process, the debit and 

 the credit are distinguished by opposite signs. 



38. The terms positive and negative, as need in the mitfao 

 matics, are merely relative. They imply that there is, either in 

 the nature of the quantities, or in their circumstances, or in the 

 purpose* which they are to answer in calculation, some such 

 opposition as requires that one should be subtracted from the 

 other. But this opposition is not that of existence and non- 

 existence, nor cf one thing greater than nothing, and another 

 kss than nothinj. For in many cases either of the signs may 

 be, indifferently and at pleasure, applied to the very same 

 quantity ; that is, the two characters may change place*. In 

 determining the progress of a ship, for instance, her easting 

 may be marked + and her westing ; or the westing may 

 be +, and the easting . All that is necessary is, that the two 

 signs be prefixed to the quantities, in such a manner as to show 

 which are to be added, and which subtracted. In different 

 processes they may be differently applied. On one conation, 

 a downward motion may be called positive, and on another 

 occasion negative. 



39. In every algebraic calculation, some one of the quantities 

 must be fixed upon to be considered positive. All other quantities 

 which will increase this must be positive also. But those which 

 will tend to diminish it, must be negative. In a mercantile 

 concern, if the stock be supposed to be positive, the profits will 

 be positive ; for they increase the stock ; they are to be added 

 to it. But the losses will bo negative ; for they diminish the 

 stock ; they are to be subtracted from it 



40. A negative quantity is frequently greater than the positive 

 one with which it is connected. But how, it may be asked, can 

 the former be subtracted from the latter ? The greater is 

 certainly not contained in the less : how then can it be taken 

 out of it ? The answer to this is, that the greater may be 

 supposed first to exhaust the less, and then to leave a remainder 

 equal to the difference between the two. If a man has in his 

 possession 1,000 pounds and has contracted a debt of 1,500; 

 the hitter subtracted from the former, not only exhausts the 

 whole of it, but leaves a balance of 500 against him. In com- 

 mon language, he is 500 pounds worse than nothing. 



41. In this way, it frequently happens, in the coarse of an 

 algebraic process, that a negative quantity is brought to stand 

 alone. It has the sign of subtraction, without being connected 

 with any other quantity, from which it is to be subtracted. 

 This denotes that a previous subtraction has left a remainder, 

 which is a part of the quantity subtracted. If the latitude of a 

 ship which is 20 degrees north of the equator is considered 

 positive, and if she sails south 25 degrees : her motion first 

 diminishes her latitude, then reduces it to nothing, and finally 

 gives her 5 degrees of south latitude. The sign prefixed to 

 the 25 degrees, is retained before the 5, to show that this is 

 what remains of the southward motion, after balancing the 20 

 degrees of north latitude. 



42. A quantity is sometimes said to be subtracted from 0. By 

 this is meant, that it belongs to the negative side of 0. But a 

 quantity is said to be added to 0, when it belongs to the 

 positive side. Thus, in speaking of the degrees of a thermometer, 

 0+6 means 6 degrees above ; and 6, 6 degrees below 0. 



AXIOMS. 



43. An AXIOM is a self-evident proposition. 



1. If the same quantity or equal quantities be add<.d to equal 

 quantities, their sums will be equal. 



2. If the same quantity or equal quantities be subtracted 

 from equal quantities, the remainders will be equal. 



3. If equal quantities be multiplied into the same, or equal 

 quantities, the products will be equal 



4. If equal quantities be divided by the same or equal quan- 

 tities, the quotients will be equal. 



5. If the same quantity be both added to and subtracted from 

 another, the value of the latter will not be altered. 



C. If a quantity be both multiplied and divided by another, 

 the value of the former will not be altered. 



7. Quantities which are respectively equal to any other qu 

 tity, are equal to each other. 



8. The whole of a quantity is greater than a part 



9. The \chole of a quantity is equal to ail its part*. 



