4.VJ 



MATH I'M \TIC8. ALGEBRA. 



[ALGEBRAIC NOTATION. 



the dividend above, and the divisor below a short lino, 



U 

 M in the notation for a fraction : thus, 12-1-4 ; 



mean the same thing; as also do a -t- b and - f> . As in 



tho furnu-r instances, so hero, tho operation can be only 

 il(J, not actually performed, except in the case of 

 number*. 



The sign stands for tho words equal to ; it is colled 

 Ike siyn of (quality : thus, 6 + 2 7 is a brief way of 

 saying that 6 plus 2 ore <<;'!/ (o 7 ; and 5 2 = 3 states, 

 in' like manner, that 5 minus 2 is e^mil to 3. Also 

 4 j- -f J JT G /-, states that four times z plus twice .T are 

 equal to six tini.-s s ; and 4 x 2 x " 2 x, expresses, 

 in symbols, that four times x minus twice x is equal to 

 twin 



A figure, or iniiiibor, pcvftud to a letter, as a mulii- 



f tli.it letter; thus, 4 is 



of x in tho quantity 4x ; and 6 is tho 



coefficient of x in Gr. In like manner, 8 is the coefficient 



<>f j ii in the quantity &ry, and 23 is the coefficient of xyz 



in tho qu.in 



Every quantity which, like each of those just noticed, 

 is not separated into parts by any plus or minus sign, is 

 called a simple quantity, or a simple expression or it is 

 said to consist of but a single term. But when a 

 quantity is made up of parts, linked together by 

 plus or minus -signs, the quantity is called a com- 



Cnd quantity, or a compound expression. The fol- 

 ing are simple expressions : they each consist of but 

 one firm. 



2ab 3abx 

 4ox, laby, lAri/3, -y, -j , Vfamnxz, <tc. 



Tho following are compound expressions : the first 

 consists of two terms, the second of three terms, and 

 the third of four terms. 



60 26, 3ax + 5by 4z, ICaxy -^ -- 2dmx + Sen. 



You see what in Algebra is meant by a term and an 

 expression ; tho first row of quantities above, is formed 

 by six distinct expressions, each expression consisting of 

 oiily one ttrm: the second row is formed by titree expres- 

 sions, of which the first consists of two terms, the second 

 of throe terms, and tho third of four terms. The first 

 row is a row of simple expressions, the second a row of 

 compound expressions. i 



We shall now give you a few easy exercises by which 

 may prove to yourself whether you fully understand 

 the meaning of the signs already explained. You must 

 not forget that whenever two or more quantities are 

 placed side by side, without any sign of operation be- 

 tween them, the multiplication of those quantities, or 

 rather of the numbers they represent, is always meant. 



EXERCISES. 



Ko.-In these exerdiei a=4, 6=2, e=S, rf^5, m=8, n=-.l. 

 All that you have to do is to give these interpretations 

 to tho letters, thus translating tho algebra into arith- 

 metic, and then to put down the numerical value of each 

 expression. To prevent mint ikes we will here show you 

 how to deal with example 4; that is, with 21m M. As 

 m U presumed to stand for 8, and d for 5, the translation 

 <.f th- exprewion is 21 X8 9x 6; that is, 108 45= 



which it the iiuim-rical value of the proposed expres- 

 sion, on tho supposition that m and d stand for 8 and 5 

 respectively. And in a similar manner are all tho other 

 examples to bo treated. Dp not forget that when tho 

 mnlti i'. :,(..!, the mx//;/.'i'i<ion 



i." interposed between them. 

 1 the values in numbers of the following expres- 

 sions : 



1. 3a-f *e* 



2. be 2a 



3. 13n-f4 



4. 21m 9rf 



5. Id + 4n 2a 



> ll Ibi 



0. 3a + 44 5c 



7. Om 5 34 



8. 14 3c-f m 



9. 114 -f n 13 

 10. 4<f-f 5m 2w 



ExtrclK* ud Quntiom In tUi Chiptcr will 



','-* 



,3. 

 ., 



2 



10 



1C. 

 17. 

 18. 



3a4 + dm 14 -f Cm 9 X 2 

 2aim 3coVn -f ^ -f arf 



Sam 2tc + * 49 

 140 



CO m -f n 



~d 



21 i aaoc ou 



m ' ~2 1" ted 



The first ten of tho?e exercises are free from 

 tho next four all contain u><j<lirai- : the four- 



teen: 1 ..'<: fraction; it is an algebraic cxpn 



consisting of but one term. Tho numerator of this 

 expression is, however, a comit>ni<l quantity, .as it is 

 niailo up of im/>fc quantities, un: - i- liy tho 



signs -f and . The leading term in <-ac.il of tho above 

 expressions, with tho exception of Exercise 10, has no 

 sign prefixed to it. You must take notice that a i 

 without any prefixed sign, is always to be regarded as 

 )ilits: the actual insertion of the -f, before a leading 

 term, is unnecessary ; since, when it is minus, the 



is always put before it. Tho plus quantities are all 

 called positive quantities, and tho minus quantities, 

 negatirc quantities. There is another thing which you 

 must also take note of. You know that the numerical 

 multi}>!i,'i- prefixed to a letter, or to a group of letters, is 

 called the cmjflcitnt of the letter or letters connected 

 with it. Now, although no coefficient should ap]>car 

 before a letter, you are not to Kay that the coefficient is 



iij, any more than you are to say, when tho plus 

 sign is absent, that the sign is nothing : the second 

 in Exercise 3, is -f 6, that is, plus once 6, and tho 

 coefficient 1, although not actually written, is to be 

 understood. If the question were asked, therefore, you 

 should say that the coefficient of the 6 is 1 : this 1 is 

 omitted from before the 6 on the same principle that the 

 -f is omitted from before the 13n ; the insertion of 

 either would add to the number of symbols, without 

 adding any clearness to the meaning of the expression ; 

 for 13 n + 6 can never be mistaken for anything but -4- 13ft 

 -f- 16, that is, thirteen times n plus b; or, as an algebraist 

 would read it, thirteen n plus b ; the word timet being 

 suppressed. Exercise 15 would be read thus : three a, 

 b ; plus d, m ; minus Jive b ; plus six c, n ; 1.1 

 nine, or 9 times 2. 



From your recollection of the terms employed in com- 

 mon arithmetic, you will know the meaning of the word 

 factor: the numbers which, multiplied together, give a 

 ''ft, are called factors of that product : so here, in 

 algebra, every multiplier is a factor: thus the factors of 

 uaj-i/ are 5, a, r, and y : tho factors of bcyz are 6, c, y, 

 and z. A product is not altered by changing the o 

 of its factors : tho following different arrangements of 

 the factors all imply the same product : bcyz, bye:, 

 <tc., <tc. ; but it is usual, w hen Ii tU-rs are the facto; 

 write them one after another in tho order in which i 

 follow in the alphabet ; so that the first of the above 

 arrangements would be that generally adopted. lu a 

 similar way, since quantities connected together, some 

 by tho additive or positive sign, and others by tho sub- 

 triK-tiee or negative sign, furnish the same result, in 

 whatever ord'-r they succeed one another, the order fixed 

 upon is mere matter of accident or choice : Exercise 10 

 is the same expression whether written 



4<i-f 5m 'Jii, or 5m 4d In, or 2n 4d + 5m; 

 but tho second of these ways would be preferred simply 



i>o a sign is thus saved. The leading term there 

 being positive, theleao. i.scd with. 



Froi i the explanation- that have now been giv. n. ami 

 from the practice afforded by tho exercises ah 

 worked, you must perceive this fact that although the 

 sign -f before an algebraic quantity is a direction to 

 add, and the sign a <lii"ction to sn'itrii-l, y. t, you 



1, in general, obey these directions till the algebra 

 is coi. :M arithmetic, by a translation of tho 



