Positive and negative quantities. Assump recognition of the use of symbols preceded by the signs and -, without any direct reference to their connection with other symbols. Thus, in the expression ab, if we are authorized to assume a to be either greater or less than b, we may replace a by the equivalent expression b + c in one case, and by b-c in the other: in the first case, we get a−b= b + c − b = b − b + c (Art. 22), b=b-c-b =0+c (Art. 16) =+c=c: and in the second a = b − b − c = 0 − c = c. The first result is recognized in Arithmetical Algebra (Art. 23): but there is no result in Arithmetical Algebra which corresponds to the second: inasmuch as it is assumed that no operation can be performed and therefore no result can be obtained, when a is less than 6, in the expression a-b (Art. 13).* 545. Symbols, preceded by the signs + or, without any connection with other symbols, are called positive and negative (Art. 32) symbols, or positive and negative quantities: such symbols are also said to be affected with the signs + and Positive symbols and the numbers which they represent, form the subjects of the operations both of Arithmetical and Symbolical Algebra: but negative symbols, whatever be the nature of the quantities which the unaffected symbols represent, belong exclusively to the province of Symbolical Algebra. 546. The following are the assumptions, upon which the tions made rules of operation in Symbolical Addition and Subtraction are lical addi- founded. in symbo tion and subtrac tion. 1st. Symbols, which are general in form, are equally general in representation and value. 2nd. The rules of the operations of addition and subtraction in Arithmetical Algebra, when applied to symbols which are general in form though restricted in value, are applied, without alteration, in Symbolical Algebra, where the symbols are general in their value as well as in their form. It will follow from this second assumption, as will be afterwards more fully shewn, that all the results of the operations of addition and subtraction in Arithmetical Algebra, will be results likewise of Symbolical Algebra, but not conversely. If we assume symbols to be capable of all values, from zero upwards, we may likewise include zero in their number: upon this assumption, the expressions a + b and a - b will become 0+b and 0-b, or + b and -- b, or b and - b respectively, when a becomes equal to zero: this is another mode of deriving the conclusion in the text. 547. Proceeding upon the assumptions made in the last Rule for Article, the rule for Symbolical Addition may be stated as follows: "Write all the addends or summands (Art. 24, Ex. 1.) in the same line, preceded by their proper signs, collecting like terms (Art. 28) into one (Art. 29): and arrange the terms of the result or sum in any order, whether alphabetical or not, which may be considered most symmetrical or most convenient." symbolical addition. terms It will be understood that negative (Art. 545) as well as positive Negative symbols or expressions may be the subjects of this operation, may occuand it is therefore not necessary, as in Arithmetical Algebra, py the first place in the that the first term of the final result should be positive (Art. 22). results of Symbolical 548. The following are examples of Symbolical Addition. Algebra. (1) Add together 3a and 5a. 3a+5a=8a (Art. 29). (2) Add together 3a and 5a. 3a-5a=-2a (Art. 31). This is exclusively a result of Symbolical Algebra. (3) Add together 3a and 5a. -3a+5a= 2a. This result, which is obtained by the Rule, is equivalent to that which would arise from the subtraction of 3a from 5a: or, in other words, the addition of 3a to 5a in Symbolical Algebra, is equivalent to the subtraction of 3a from 5a in Arithmetical Algebra. (4) Add together 3a and -5a. -3a-5a=-8a (Art. 31). This is exclusively a result of Symbolical Algebra: in contrasting it however with Ex. 1, it merely differs from it in the use of the sign throughout, instead of the sign +. Examples. - 4 a - 4x3 Rule for In these examples, the coefficients of the like terms, which have the same sign, are added together, and to the difference of the sums, preceded by the sign of the greater, is subjoined the symbolical part of the several like terms: it is the rule given in Art. 31, applied without any reference to the signs of the first term. (8) 3a-4b (9) -7x2+6xy- 7y2 In these examples, the sets of like terms are severally combined into one (Art. 31), and arranged, in the result, in alphabetical order, no regard being paid to the placing a positive term, when any exists, in the first place. The several sets of like terms are collected together out of the several addends. The alphabetical order of the symbols is, in this case, reversed. It should be kept in mind in this and in all other cases that the arrangement of the terms in the final result, does not affect its value or signification, but is merely adopted as an aid to the eye or to the memory, or with reference to peculiar circumstances connected with some one or more of the symbols involved: see the Examples in Art. 33. 549. The rule for subtraction in Symbolical Algebra is in Symbo- derived, by virtue of the assumptions in Art. 546, from the lical Alge- corresponding rule in Arithmetical Algebra: it may be stated bra. as follows. To the minuend or minuends, add (Art. 547) the several terms of the subtrahend or subtrahends with their signs changed from to and from to +." The following are examples of Symbolical subtraction. (1) From a subtract - b. The result is a+b: or the symbolical difference of a and -b is equivalent to the sum of a and b. (2) 7a (3) -7a 3 a 4a (4) 7 a (5) 7 a In these examples, the minuend and subtrahend are written underneath each other, as in common Arithmetic: the results are severally the same as in the following examples of addition. These examples are respectively equivalent to (9)-a-b-(a - b), or a-b-a+b=-2a. The terms of the several subtrahends are included between brackets, and, when the brackets are removed, all their signs are changed (Art. 24). (10) a3+3a2x + 3 ax2+x3 a3-3 a*x + 3a x2 – x3 6a2x+2x3 (11) 3a-4b+ 7c- 9d 2b-10c 6d +14e 3a6b+17c- 3d-14e Examples. The same terms are used in Arithmetical and (12) From 3x-7y subtract x + 2y and −7x+4y. 3x-7y-(x+2y) − (−7x+4y) The several subtrahends included between brackets and preceded by the sign are written in the same line with the minuend; the brackets are subsequently removed and the signs of the several terms which they include changed, in conformity with the Rule. (13) x2+2xy-y3 − {x2 + xy − y2 + (2 x y − x3 — y3)} = x2 + 2 x y − y2 — x2 − x y + y2 − (2 x y − x2 — y3) = xy-2xy + x2+y3 In this case, we first remove the exterior brackets and reduce to their most simple equivalent form the terms which are external to those which remain: we then remove the remaining brackets and arrange the terms, when reduced, in alphabetical order (Arts. 20 and 21.) (14) a-[a+b− {a + b + c − (a+b+c+d)}] In this case, we have a triple set of brackets which are successively removed, and the like terms, which they involve, are obliterated or reduced into one. See the Examples in Art. 33. 550. In the exposition and exemplification of the preceding rules, we have felt it to be unnecessary to repeat definitions and assumptions which are common to Symbolical and Arithmetical Symbolical Algebra: such are the ordinary uses and meanings of the signs Algebra. In Arith- + and -, of coefficients (Art. 25), of like and unlike terms (Art. 28), metical of indices and powers (Arts. 38 and 39), and the methods of Algebra, the defini- denoting the ordinary operations (Art. 9). tions deter mine the rules of 551. The use however, of the same terms in these two sciences operation: will by no means imply that they possess the same meaning in all in Symbo- their applications. In Arithmetic and Arithmetical Algebra, adlical Algedition and subtraction are defined or understood in their ordinary sense, and the rules of operation are deduced from the definidetermine tions: in Symbolical Algebra, we adopt the rules of operation bra the rules of operation |