Page:Encyclopædia Britannica, Ninth Edition, v. 1.djvu/558

Rh 520 A L G E B K A [FUNDAMENTAL Write down the expression in x by putting - for y, &c. X It becomes -- . z #- 2 x&quot; = 2 because x = 0. SECT. I. FUNDAMENTAL OPERATIONS. The primary operations in algebra are the same as in common arithmetic namely, addition, subtraction, multi plication, and division ; and from the various combinations of these four, all the others are derived. I. Addition. 10. In addition there maybe three cases: the quanti ties to be added may be like, and have like signs ; or they may be like, and have unlike signs ; or, lastly, they may be unlike. Case 1. To add quantities which are like, and have like signs. Rule. Add together the coefficients of the quantities, pre fix the common sign to the sum, and annex the letter or letters common to each term. Add together EXAMPLES. + 7a + 3a ) + a (+ 2a Add together Sum, +13a Sum, - 20ax Case 2. To add quantities which are like, but have unlike signs. Ride. Add the positive coefficients into one sum, and the negative ones into another; then subtract the less of these sums from the greater, prefix the sign of the greater to the remainder and annex the common letter or letters as before Add together Sum of the pos. Sum of the neg. EXAMPLES. + lax &quot;* Add together- OCIX + Sax + 6a&+ 7 - 4a6 + 9 + ab- 5 + &amp;gt;7a6-13 + ax Sum of the pos. + 14a Sum of the neg. - 4a6 IS Sum required, + 7 ax Sum required, + 10a6- 2 Case 3. To add unlike quantities. Rule. Put down the quantities, one after another, in any order, with their signs and coefficients prefixed. EXAMPLES. 2a 36 -4c ax + lay bb - 3bz Sum, ax + 2ay + bb 36z Sum, 2a + 36-.4c IT. Subtraction. 1 1 . General Rule. Change the signs of the quantities to be subtracted, or suppose them changed, and then add them to the other quantities, agreeably to the rules of addition. EXAMPLES From 5a-126 Subtract 2a 56 Remainder 3a - 76 5xy - 2 + Sx - y 3xy 8 Sx 3y From Gz- Sy + 3 Subtract 2x + 9 ?/ - 2 Remainder 4# - 1 7y + 5 aa a.r - yy bb -by + zz 2xy + 6 + 1 Qx + 2y aa ax yy bb + by zz The reason of the rule for subtraction may be explained thus. Let it be required to subtract 1p - 3q from m + n. If we subtract 2p from m + n, there will remain m + n - 1r&amp;gt;, but if we are to subtract 2p - 3q, which is less than 2p, it is evident that the remainder will be greater by a quantity equal to 3q; that is, the remainder will be m + n - 2v + 3q, hence the reason of the rule is evident III. Multiplication, 12. General Rule for the Signs. If the qiiantities to be Rule of multiplied have like signs, the sign of the product is + ; signs. but if they have unlike signs, the sign of the product is. This rule, which is given by Diophantus 1 as the defini tion of + and, may be said to constitute the basis of algebra as distinct from arithmetic. If we admit the definitions given above, the rule may be demonstrated in the following way : (1.) +ax+6=+a6is assumed. (2.) + a x b will have the same value, whatever 6 may be connected with, as it has when - 6 is connected with +6 (Law 1). Now + ax( + b-b) = + a x + = (Def.) But + ax( + b-b}=+ay. + b, and + a x -b (Law 2). . . + a x + b and + a x -b make up i.e., + ab and -fax - b make up 0. Now + ab - ab 0, . . + a x b= - ab. (3.) Similarly -ax -5= +ab. The examples of multiplication may be referred to two cases ; the first is when both the quantities are simple, and the second when one or both of them are compound. Case 1. To multiply simple quantities, Ride. Find the sign of the product by the general rule, and annex to it the product of the numeral coefficients; then set down all the letters, one after another, as in one word. EXAMPLES. / Multiply + a 1. J B ? V Product + ac + 56 - 4a -20a6 3. - 2laabx Case 2. To multiply compound quantities. Rule. Multiply every term of the multiplicand by all the terms of the multiplier, one after another, according to the preceding rule, and collect their products into one sum, which will be the product required. 1. Mult. 2x +y By x -2y EXAMPLES. 2. a -6 +c a + b c Prod. 2 2xx - 3xy 2yy aa- ab + ac + ab -bb + bc - ac + be - cc aa -bb + 2bc-cc weu.(tt n&amp;gt;n7 &amp;gt;. .7&amp;lt;J,it. Diopliantus, Ed. Fenrutt, Tolosce, 1070, p. 7, Def. 9.