A L G 



A L G 



In algebra, as \vc have already ilattd, evjry quantity, 

 whether it be known or given, or unknown or required, is 

 ufually reprtfentcd by fomc letter of the alphabet ; and the 

 given quantities are commonly denoted by the initial letters, 

 <7, i, c, d, kc. and the unknown ones by the final letters, 

 V, w, X, y. Thefe quantities are connected together by 

 ccitain figns or fymbols, which fcrve to (licw ihcir mutual 

 relation, and at the fame time to finiplify the fv ienee and to 

 reduce its operations into a lefs compafs. Accordingly the 

 fign -|- /'^"■' "'■ """■'■, fignifies that the quantity, to wliich it 

 is prefixed, is to be added, and it is called a pofitive or affir- 

 mative quantity. Thus, a -^ 1/ exprtfTes the fuin of the two 

 quantities a and /', fo that if a were 5, and /, 3, a -j- * 

 would be 5 + 3, or S. If a quantity have no fign, -\- plus 

 is underftood, and the quantity is affirmative or pofitive. 

 The fign — , nilnus or lefs, denotes that the quantity which 

 it precedes is to be fubtracled, and it is called a negative 

 qiiantity. Thus a ~ b exprefle'; the difference uf u and L : 

 fo that a being 5, and h, 3, « — i or 5 — 3 would be equal 

 to 2. If more quantities than two were connefttdby thel'e 

 figns, the fum of thofe with the fign — mull be fubiradlcd 

 from the fum of tho(e: with the fign -j-. Thus, a — I -\- 

 c — </reprefents the quantity which would remain, when c 

 and d are taken from a and v. So that if a were 7, b, 6, 

 c, 5, and <7, 3, rt + i — r — r/ or 7 + 6 — 5 — 3, or 1 3 — 

 8, would be equal to 5. If two quantities are coimecled 

 by the fign m , as a v: b, this mode of expreffion rcprcfents 

 the difference of a and b, when it is nol known which of 

 them is the greatefl. - 



The fign X fignifies that the quantities between wliich it 

 ftands are to be multiplied together, or it i-eprcfcnts their 

 produft. Thus, a X b exprcffes the produ^S of a and b ; 



a X b X c denotes the produft of a, b, and c ; ^ -]- b x c 

 denotes the produft of the compound quantity a -{■ b h\ the 



fimple quantity c ; and a -^ b -\- c x a — b + r x a -\- b 

 reprefents the produft ofthe three compoundquantities,multi- 

 plied continuallyinto one another; fo that if ;! were y, 1^,4, and 



C, 3, then wotild a -{• b -\- c X a — b -{■ c x a + c be 

 I2X 4 X 8, or 384. The line conue(5ling the fimple quan- 

 titiei and forming a compound one, placed over them, is 

 called a vinculum. Qiiantities that are joined together 

 without any intermediate fign form a product ; thus a bh 

 the fame with a X b, and ale the fame with a X b X c. 

 When a quantity is multiphed into itfelf, or raifed to any 

 pow-er, the ufual mode of exprefPion is to draw a line over 

 the quantity and to place the number denoting the power at 

 the end of it, which number is called the index or exponent. 



Thus, 



a -\- by denotes the fame as a -f- i x a + ^ or fe- 

 cond power, or fqnare, oi a -\- b confidered as one quan- 

 tity ; and a -\- t\ denotes the fame asrt+i5x a + b x 



a + b, or the third power, or cube, oi a + b. In exprcf- 

 fing the powers of quantities reprcfented by fingle letters, 

 the line over the top is ufually omitted ; thus, a' is the fame 

 as a a or fl X a, and P the fame a^ b b b or b X b X b, and 

 a' b', the fame as a a b b b ov a X a X b X b X b. The full 

 point . and the word into, are fometimes ufed inllead of x , as 



Thus, 



fion ; thus, a -^ /) n b, denotes that a i is to be divided by 

 a + b. But the divifion of algebraic quantities is molt 

 commonly eNiirelfed by placing the divifor under the divi- 

 dend with a line between thtm, likea vulgar fradftion. Thus, 



-reprefents the quantity arifing by dividing c by b, or the 



a + b 



quotient, and 



a + c 



reprefents the quotient of a + b di 



/I 



b + be 



would be 



J^ 



the fign of multiplication. 



a + b . a + 



and 



a -\- b into a ■{■ c, fignify the fame thing as a + i X a + f, 



cr the produft of « + 3 by a + c. 



The fign — is the fign of divifion, as it denotes that the 

 quantity preceding it is to be divided by the fucceedlng 

 quantity. Thus, c-— b fignifies that c is to be divided by 



i ; and a -\- b ~ a + c, that a + b is to be divided by 



« + f . The mark ) is fometimes ufed as a note of divi- 



\ldcd bv a + c. Quantities thus exprefTed are called alge- 

 braic friiftions. See Fraction. 



The fign V^exprelTes the fquare root of any quantity to 

 which it is prefixed ; thus V 25 fignifies the fquare root of 

 25 or 5, becaufe 5 X 5 is 25 ; and V a i denotes the fquare 

 rootufai;and j ab + i ^ denotes the fquare root of 



V 7i 



— '■ , or of the quantity arifing from the divifion of 



// 



(lb + be by J; but s'Tb +'b e, which has the feparating 



J 

 line drawn under v'~ fignifies that the fquare root 0/ 

 ab + be is to be /// taken, and afterwar ds divided b y J ; 

 fo that if a were 2, /., C, c, 4, and d, 9, -^ a b ■^- b e w ould 



d 



be iS or-; but 



.9. 9 



which is 2. 



The fign ^/ with a figure over it is ufed to exprcfs the 

 cubic or biquadratic root, &c. of any quantity ; thus y^' C4 

 reprefents the cube root of 64 or 4, becaafe 4 x 4 X 4 is 

 64 ; and \/ ba + c d the cube root of a b + e d. In like 

 manner y/TS denotes the biquadratic root of 16, or 2, 

 becaufe 2x2x2x2 is 1 6, and y^/ai + c d denotes 

 the biquadratic root o{ ab + e d ; and fo of others. Quan- 

 tities thus expreffed are called radical qtiantities, or Sur ds ; 

 of which thofe, confifting of one terra only, as v'' " and 

 ^ a b, are called fimple furds ; and thofe confiding of fevenJ 



terms, or numbers, as v''''"~^and ■\/a- — b--\-hc, are de- 

 nominated compound furds. Another commodious method 

 of expreiring ra<!ie:il quantities is that which denotes the 

 root by a \ulgar fraftlon, placed at the end of a hue drawn 

 over the quantity given. In this notation, the fquare root 



is expreffed bv , the cube root by - , the biquadratic ^oot 



■ ^ 3 



by ~, &c. Thus a 1^ expreffcs the fame quantity with 



a/ a /. e. the fqiiarc root of a, and a' + a /-Inhefameas 



^ / a^ + ab, 1. e. the cube root of .i' + a b ; a!id a I -• dt- 

 notes the cube root of the fquare of a or the fquare ofthe cube 

 root of a ; and a + s;!* the feventh power of the biqua- 

 dratic root of a + s : and fo of others ; a' \i is a, a^\i 

 is a, 5(c. When the root of a quantity reprefented by a 

 fimple letter is to be expreffed, the line over it may be 

 omitted; fo that ai fignifies the fame as a |:, and bl the fame 



asTjj or \/'^- Quantities that have no radical fign (^/) 

 or index annexed to them, arc called rational quantities. 

 The fign =, called the fign of equality, fignifiti that 



the 



