MUEJSKEN
A L GE B

R A.

401

Fundamen- of geometry and algebra, there may be an oppofition terms in the quantities to be added may be united, fo Fundamental opera- or contrariety in the quantities, analogous to that of as to render the expreffion of the fum more fimple.

tions.

addition and subtraction; and the figns + and
very conveniently be used to exprefs that contrariety.
may
In fuch cafes, negative quantities are understood to
exift by themselves; and the fame rules take place in
operations into which they enter, as are used with regard
to the negative terms of abftra& quantities.

CH, A P. I.

SECT. I. Fundamental Operations."

THE fundamental operations in algebra are the fame as in common arithmetic, Addition, Subtraction, Multiplication, and Divifion; and from the various combinations of these four, all the others are derived.

PROB. I. To add Quantities.

Simple quantities, or the terms of compound quantities,
to be added together, may be like with like figns, like
with unlike figns, or they may be unlike.

Cafe 1. To add terms that are like and have like
figns.

Rule. Add together the coefficients, to their fum pre-
fix the common fign, and subjoin the common letter
or letters.

Examp. To 5ab

3aa-ab

Add 4ab

7aa-zab

4na-5ab

Sum 9ab

14aa-8ab.

PROB. II. To fubtract Quantities

General Rule. Change the figns of the quantity to be
fubtracted into the contrary figns, and then add it,
fo changed, to the quantity from which it was to be
fubtracted (by Prob. f.); the fum, arifing by this ad-
dition is the remainder.

Examp. From

+5a, Subtract +3a

Rem. +2a

7ab-16bc
3ab+mb

4ab-16bc-mb

From 54-76 +90 +8

Subt. 2a-4b+9c-d

When a positive quantity is to be fubtracted, the rule is obvious from Def. 3. In order to fhow it, when the negative part of a quantity is to be fubtracted, let to the rule, is a-c+d. For if is fubtracted from c-d be fubtracted from a, the remainder, according fmall, because c is fubtracted inftead of c-d, which is a, the remainder is a-c (by Def. 3.); but this is too lefs than it by d; the remainder therefore is too fmall by d; and d being added, it is a-c+d, according to the rule.

Other wife, If the quantity d be added to thefe two quantities a and c-d, the difference will continue the fame; that is, the excefs of a above c-d is equal to the excefs of a+d above c―d+d, that is, to the excefs of a+d above c, which plainly is a+d-c, and is

Cafe 2. To add terms that are like, but have unlike therefore the remainder required. figns.

Rule. Subtract the lefs coefficient from the greater; prefix the fign of the greater to the remainder, and fubjoin the common letter or letters.

The reafon of this rule is derived from Def. 6. and
addition of one of the quantities to be multiplied as
from the nature of multiplication, which is a repeated
often as there are units in the other.
letters in two terms multiplied together may be placed
in any order, and therefore the order of the alphabet
is generally preferred.

Cafe 2. To multiply compound quantities.
Rule. Multiply every term of the multiplicand by all
the terms of the multiplier, one after another, accord-
ing to the preceding rule, and then collect all the
products into one fum; that fum is the product re-
quired.

BIBLIOTHECA

3 E

Examp.

tal operations.

Fundamen- Examp. Mult. 2a+36

tal opera-
tions.

By 3ax-4by

m+x
m-x

mm+mx

-N:XXX

mm-xx

Prod. ac-cb-ad+db

Of the general Rule for the Signs.

The reafon of that rule will appear by proving it, as applied to the laft mentioned example of a-b multiplied by c-d, in which every case of it occurs.

ber of terms of a compound quantity, to denote thofe Fundamen which are understood to be affected by the particular tal opera fign connected with it.

Thus, in the laft example, it fhows that the terms +a and b, and alfo c and d are all affected by the fign (X). Without the vinculum, the expreffion a-bxc-d would mean the excefs of a above be and d; and a-bxc-d would mean the excess of the product of a−b by c, above d. Thus alfo, a+b) expreffes the fecond power of ath, of the product of that quantity multiplied by itself; whereas a+b would exprefs only the fum of a and b'; and fo of others. By fome writers a parenthesis () is ufed as a vincu‣ lum, and (a+b)' is the fame thing as a+b).

Since multiplication is a repeated addition of the multiplicand as often as there are units in the multiplier, hence, if a-b is to be multiplied by c, a-b must be added to itself as often as there are units in c, and the product therefore must be ca-cb (Prob. I.). But this product is too great; for a-b is to be multiplied, not by c, but by c-d only, which is the excefs of c above d; d times a- -b therefore, or da-db, has been taken too much; hence this quantity must be fubtracted from the former part of the product, and the remainder, which (by Prob. II.) is ca-cb-da+db, will be the true product required.

Def. 12. When feveral quantities are multiplied together, any of them is called a factor of the product.

13. The products arising from the continual multiplication of the fame quantity are called the powers of that quantity, which is the root. Thus, aa, aaa, aaaa, &c. are powers of the root a.

14. Thefe powers are expreffed, by placing above the root, to the right hand, a figure, denoting how often the root is repeated. This figure is called an index, or exponent, and from it the power is denominated. Thus,

PROB. IV. To divide Quantities.

tions.

General Rule for the Signs. If the figns of the divifor and dividend are like, the fign of the quotient is +; if they are unlike, the fign of the quotient is.

This rule is eafily deduced from that given in Prob. III.; for, from the nature of divifion, the quotient must be fuch a quantity as, multiplied by the divifor, fhall produce the dividend with its proper fign.

The 2d and 3d powers are generally called the fquare and cube; and the 4th, 5th, and 6th, are alfo fometimes refpectively called the biquadrate, furfolid,

and cubocube.

From Def. 8. the quotient of any two quantities. may be expreffed, by placing the dividend above a line and the divifor below it. But a quotient may often be expreffed in a more fimple and convenient form, as will appear from the following diftinction of the

Cor. Powers of the fame root are multiplied by adding their exponents. Thus, a Xa3=a3, or aaax aaaaaaa, b3Xb=ba.

Scholium.

cafes.

Cafe I. When the divifor is fimple, and is a factor of all the terms of the dividend. This is eafily dif covered by inspection; for then the co-efficent of the divifor measures that of all the terms of the dividend, and all the letters of the divifor are found in every term of the dividend.

Sometimes it is convenient to exprefs the multiplication of quantities, by fetting them down with the fign (X) between them, without performing the operation according to the preceding rules; thus a'Xb is written inftead of ab; and a-bxc-d expreffes the product of a-b, multiplied by c-d.

Rule. The letter or letters in the divifor are to be expunged out of each term in the dividend, and the coefficients of each term to be divided by the coefficient of the divifor; the quantity refulting is the quotient.

Def. 15. A vinculum is a line drawn over any num

Ex. a) ab(b. 2aab) 6a3bc-4a'bdm (zac—2dm.

The reafon of this is evident from the nature of divifion, and from Def. 6. Note. It is obvious from corollary to Prob. III. that powers of the fame root are divided by fubracting their exponents.

Thus a3)a3 (a_a3)a1 (aa.

Thus a')a3 (a a3)a' (aa. Also a3b)a3b6 (ab3.