2. Again, suppose that it is required to subtract 5-3 from 9 If we subtract 5 from 9, the remainder will be 9-5; but the quantity to be subtracted is 3 less than 5, and we have, therefore, subtracted 3 too much; we must, therefore, add 3 to 9~5, which gives for the true remainder, 9-5+3, which is equal to 7. 3. Let it now be required to subtract b-c from a. If we take b from a, the remainder is a-b; but, in doing this, we have subtracted c too much; hence, to obtain the true result, we must add c. This gives, for the true remainder, a—b+c. If a=9, b=5, and c=3, the operation and illustration by figures would stand thus: from a from 9 Rem. 9-5+3 -9 =2 =7 The same principle may be further illustrated by the following examples. 4. a-(c-a) a-c+a=2a-c. a—(a—c)=a—a+c=c. a+b—(a—b) =a+b—a+b=26. Let it be noted, that in the result in each of the preceding examples, the signs of the quantity to be subtracted have been changed from plus to minus, and from minus to plus; hence, in order to subtract a quantity, it is merely necessary to change the signs and add it. Hence, the RULE, FOR FINDING THE DIFFERENCE BETWEEN TWO ALGEBRAIC QUANTITIES. Write the quantity to be subtracted under that from which it is to be taken, placing similar terms under each other. Conceive the signs of all the terms of the subtrahend to be changed, and then reduce the result to its simplest form. NOTE. It is a good plan with beginners, to direct them to write the example a second time, and then actually change the signs, and add, as in the following example. They should do this, however, only till they become familiar with the rule. From 5a+3b-c The same, with the Take 2a-2b-3c signs of the subtraRemain. 3a+5b+2c hend changed. 5a+3b-c -2a+2b+3c 3a+5b+2c 11. 3a--2b 6ax-4y2+3 5a-3b 3ax-6y2+2 . Ans. 19-ab. Ans. b. Ans. -b. Ans. 5. Ans. ax-7. Ans. 2y. Ans. 2y. Ans. 2x-2y. Ans. 2y+2z. Ans. x-2z. Ans. 2a. Ans. 11a. Ans. 5a. Ans. -6b. Ans. a+b. Ans. 3a+2b. Ans. -12a. Ans. 0. Ans. -3a+5b, or 5b-3a. Ans. 16. Ans. 7. Ans. 20. Ans. a+5b+9. Ans.-9. 37. From 3a-2b+6, take 2a-7b--3. Ans. 5a+4b+7d-12. 39. From 7a+3m-8x, take -6a-5m-2x+3d. Ans. -a+8m-6x-3d. 40. From 32a+3b, take 5a+17b. 41. From 6a+5-3b, take -2a-9b-8. 42. From 3c-21+5c, take 81+7c-4l. 43. From 3ax-2y2, take -5ax-8y2. . 44. From 2x2-3a2x2+9, take x2+5a2x2-3. Ans. x2—8u2x2+-12. 45. From 4x2y3—5cz+-8m, take —cz+2x2y3-4cz. 46. From 11xyz+3a, take -6xyz+7-2a-5xyz. 48. From 3a(x-z), take a(x—z). . Ans. 2a(x-2). 49. From 7a2(c-z)—ab(c—d), take 5a2(c—z)—5ab(c–d). Ans. 2a2(c-z)+4ab(c--d). ART. 59. It is sometimes convenient to indicate the subtraction of a polynomial without actually performing the operation. This may be done, if it is a monomial, by placing the sign minus before it; and, if it is a polynomial, by enclosing it in a parenthesis, and then placing the sign minus before it. Thus, to subtract a-b from 2a, we may write it 2a—(a—b), which reduces to a+b. By this transformation, the same polynomial may be written in several different forms; thus: a—b+c—d—a—b—(d—c)—a—d▬—(b—c)=a—(b−c+d). Let the pupil, in each of the following examples, introduce all the quantities, except the first, into a parenthesis, and precede it by the sign minus, without altering the value of the expression. It will be found a useful exercise for the pupil, to take each of the preceding polynomials, and without changing their values, write them in all possible modes, by including either two or more terms in a parenthesis. OBSERVATIONS ON ADDITION AND SUBTRACTION. ART. 60. It has been shown, that Algebraic Addition is the process of collecting, into one, the quantities contained in two or more expressions. The pupil has already learned, that these expressions may be all positive, or all negative, or partly positive and partly negative. If they are either all positive, or all negative, the sum will be greater than either of the individual quantities; but, if some of the quantities are positive and others negative, the aggregate may be less than either of them, or, it may even be REVIEW. In subtracting b-c from a, after taking away b, have we subtracted too much, or too little? What must be added, to obtain the true result? Why? What is the general rule for finding the difference between two algebraic quantities? 59. How can the subtraction of an algebraic quantity be indicated? nothing. Thus, the sum of +4a and -3a is a; while that of +a and -a, is zero, or 0. As the pupil should have clear views of the use and meaning of the various expressions employed, it may be asked, what idea is he to attach to the operations of algebraic addition and subtraction. ART. 61. In common or arithmetical addition, when we say, that the sum of 5 and 3 is 8, we mean, that their sum is 8 greater than 0. In algebra, when we say that 5 and -3 are equal to 2, we mean, that the aggregate effect of adding 5 and subtracting 3, is the same as that of adding 2. When we say, that the sum of -5 and +3, is −2, we mean, that the result of subtracting 5, and adding 3, is the same as that of subtracting 2. Some algebraists say, that numbers with a positive sign represent quantities greater than 0, while those with a negative sign, such as -3, represent quantities less than nothing. The phrase, less than nothing, however, can not convey an intelligible idea, with any signification that would be attached to it in the ordinary use of language; but, if we are to understand by it, that any negative quantity, when added to a positive quantity, will produce a result less than if nothing had been added to it; or, that a negative quantity, when subtracted from a positive quantity, will produce a result greater than if nothing had been taken from it, then the phrase has a correct meaning. The idea, however, would be properly expressed, by saying, that negative quantities are relatively less than zero. Thus, if we take any number, for instance 10, and add to it the numbers 3, 2, 1, 0, −1, −2, and 3, we see, that adding a negative number produces a less result than adding zero. From this, we also see, that adding a negative number, produces the same result, as subtracting an equal positive number. Again, if we take any number, for example 10, and subtract from it the numbers 3, 2, 1, 0, −1, −2, and 3, we see, that subtracting a negative number produces a greater result than subtracting zero: From this, we also see, that subtracting a negative number, produces the same result, as adding an equal positive number. REVIEW.-60. When is the sum of two algebraic quantities less than either of them? When is the sum equal to zero? ART. 62. In consequence of the results they produce, it is customary to say, of two negative algebraic quantities, that the least is that which contains the greatest number of units. Thus, -3 is said to be less than -2. But, of two negative quantities, that which contains the greatest number of units is said to be numerically the greatest; thus, -3 is numerically greater than -2. ART. 63. A correct idea of the.nature of the addition of positive and negative quantities, may be gained by the consideration of such questions as the following: Suppose the sums of money put into a drawer to be positive quantities, and those taken out to be negative; how will the money in the drawer be affected, if, in one day, there are 20 dollars taken out, afterwards 15 dollars put in, after this 8 dollars taken out, and then 10 dollars put in? Or, in other words, what is the sum of -20, +15, −8, and +10? The answer, evidently, is -3; that is, the result of the whole operation diminishes the amount of money in the drawer 3 dollars. Had the sum of the quantities been positive, the result of the operation would have been, an increase of the amount of money in the drawer. Again, suppose latitude north of the equator to be reckoned +, and that south, ; and that the degrees over which a ship sails north, are designated by +, while those she sails over south, are designated by, and that we have the following question: A ship, in latitude 10 degrees north, sails 5 degrees south, then 7 degrees north, then 9 degrees south, and then 3 degrees north; what is her present latitude? This question is the same as to find the sum of the quantities +10, −5, +7,−9, and +3; this is evidently +6; that is, the ship is in 6 degrees north latitude. Had the sum of the negative numbers been the greater, it follows, that the ship would have been found in south latitude. Other questions of a similar nature may be used by the instructor, to illustrate the subject. ART. 64. Subtraction, in arithmetic, shows the method of finding the excess of one quantity over another of the same kind. In this case, the number to be subtracted must be less than that from which it is to be taken; and, as they are considered without refer REVIEW.-61. What is meant, by saying that the sum of +5 and −3, is equal to +2? What is meant, by saying that the sum of 5 and 3, is equal to -2? Is it correct to say, that any quantity is less than nothing? What is the effect of adding a positive quantity? Of adding a negative quantity? Of subtracting a positive quantity? Of subtracting a negative quantity? 62. In comparing two negative algebraic quantities, which is called the least? Which is numerically the greatest? |