37. To add quantities which are alike, and have unlike signs. RULE I. Add all the affirmative coefficients into one sum, and the negative ones into another. II. Subtract the less of these sums from the greater, to the remainder prefix the sign of the greater, and subjoin the common letter or letters, as in the preceding rule f. f In order to explain this rule, it will be necessary to suppose a case. A trader in settling his books finds that several sums are due to him, and that he is indebted in several sums to others; to the former of these, (they being additive, or tending to increase his stock,) he prefixes for the sake of distinction the sign+, and to the latter, (these being subductive, or tending to decrease his stock,) he prefixes the sign Now in adjusting the account, he must add all the sums owing to him into one sum, and the sums he owes into another; he must then subtract the less of these sums from the greater, and the remainder will truly shew what he is worth. If the amount due to him be the greater, the remainder must have the sign + prefixed, and shews that he has property to that amount; if the sum of his debts be the greater, then the remainder will have the sign -, and shew that he is so much in debt. Now to apply this doctrine, (which is evident to the meanest capacity,) let us take the 12th example, where if a be considered as representing a pound sterling, +7x will be 7 pounds, and +8x eight pounds, both sums due to him; wherefore -3x and -2x will represent 3 pounds and 2 pounds, both sums due by him; now the sum of the two former, (+7x and + 8 x,) or 15x, viz. 15 pounds, will be his property, and the sum of the two latter, (−3≈ and 2x,) or 5a, will be the sum of his debts, viz. 5 pounds; he therefore is worth 15 pounds, but has to pay 5 pounds out of this sum; the balance of his account then will be (15 - 5x, or) 10 x, or ten pounds, as in the example. ing 12. Add the following algebraic quantities together. OPERATION. Explanation. I first add the affirmative quantities +7x, and + 8 ≈ together, making + 15; I then add the negative quantities -3x and 2 x together, making -5; next I subtract 5 from 15, and to the remainder 10 prefix the sign + of the greater sum, and subjoin the common letter л, mak10x for the sum: the sign + might have been omit ted. 38. To add unlike quantities together", RULE I. Place like quantities under one another in a column, That algebraic addition should sometimes require addition, and sometimes subtraction, appears at first sight an unaccountable paradox, but on an attentive examination the paradox vanishes; "it arises wholly from the scantiness of the name, from employing an old term in a new and enlarged sense :" if instead of addition you call this process incorporation, striking a balance, union, or any name sufficiently extensive to convey an adequate idea of the two-fold operation employed, the difficulty will be removed. ́s In this example, if 22 be supposed to represent a pound sterling, and if + be prefixed to the sums due to, and to the sums due by any person, then if the example be considered as a true statement of his accounts, it appears that he is worth (-2022 or) -20 pounds; that is, he is 20 pounds in debt. Like quantities may be incorporated together, but unlike quantities cannot; we can add 2 pounds and 3 pounds together, and the sum is 5 pounds; but we cannot add 2 pounds and 3 shillings together, that is, we cannot incorporate them, so as to make either 5 pounds or 5 shillings: all that can be done is to write them down one after the other, thus, 2 pounds + 3 shillings; or in the usual way, 21. 3s. The same holds true of algebraic quant it ies; we cannot add b to a, so as to make either 2a or 26; all that can be done is to each with its proper sign; and there will be as many columns as there are different kinds of quantities. II. Add each column separately, and if the quantities in any column have like signs, add it by Art. 36. but if unlike, add it by Art. 37: the results, with their proper signs, placed in a line below, will be the sum required. Note. It is usual to begin at the left hand column, and proceed from thence, in order, to the right, 23. Add the following algebraic quantities together; 2 a + 3b-4c5a + 2 b − 7 c + 3 a 2 b + 8c + 2a + b OPERATION. 2a+3b-4 c 5a+2b 7 c За 2b+8c -7 C Explanation. - 7 c. I place all the a's in the first column, and because they are all, and likewise the leading quantities, I omit the sign; I then place all the 's with their proper signs in the second column, and all the c's in the third; I next add up the first, or left hand column, by Art. 36, and place the sum 12a below: I then add the second and third columns by Art. 37, the sum of the former being + 4b, and that of the 10c; these three sums placed in a line, with their proper signs, are the answer. 2a + b 12 a+ 4 b latter 10 c place them one after the other, each with its proper sign, thus a+b. And when there are several quantities concerned, to collect all that are of one kind into one sum, and all those of another kind into another, &c. by the former rules, and then write down the several sums in succession, each with its proper sign. · x + y z together. Sum 2x + 3y - 6 z. 7b8a 9 b + 2 a − 2b to 8z+ 2x + 3y + z -bcd8 together. Sum 6a - 3 bc + d - 8. 39. To add quantities under a vinculum1. RULE. I. If the quantities under the vinculum are alike, viz. if they consist of the same letters, numbers, coefficients, signs, and indices, they are added by taking the sum of the coefficients without the vinculum, and subjoining the vinculum, with the quantities under it, to this sum, II. But if the quantities under the vinculum are unlike, (viz. i The vinculum connects all the quantities included under it into one, (Art. 26.) they are consequently to be managed collectively like a simple quantity. When the quantities under the vinculum are in all respects alike, (namely, in signs, letters, and indices,) the addition will be performed by adding the coefficients, (or numbers connected with, but not under, the vinculum,) and subjoining the common vinculum to the sum, as is shewn in Art. 36. 37. But when the quantities under the vinculum are not in all respects alike, (namely, if they differ either in the signs, letters, or indices,) they are evidently unlike quantities, and admit of no other addition than merely connecting them by the signs of their coefficients, as in Art. 38. if they differ in any of those particulars,) they can only be connected together by their proper signs. 37. Add 3x + y + 4 √ x + y + 5 √ x + y + √x + y together. OPERATION. 3√x x + y 4 √x + y 5x+y Sum 13 x + y Explanation. The quantities under the vinculum being alike in all respects, I merely add the coefficients 3, 4, 5, and 1, (understood,) together, by Art. 36. and to their sum 13 subjoin the vinculum for the answer. 40. Add 2a + 3 √ x + 1 + 2a + 4 √x + 1 +3a-2 √x + 1 together. Sum 6 a + 6 √x + 1. 41. Add ax + 2b 2 − b + a + b } } } − 8 b 2 − 3 b + 2.ab +3.a+b. together. 3 ax + b2 Sum 10 ax 42. Add a/b + √x + y + z + 4 √ x + xyz √x + y + z − 2 √a + 2a√b together. y -4b+ 12 ax 3 a √b + Sum 4 √x + y |
