CHA P. L

Fundamental Operations.

'HE fundamental operations in Algebra are the fame as in common Arithmetic, ADDITION, SUBTRACTION, MULTIPLICATION, and DIVISION; 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.

CASE I. To add Terms that are like, and have like figns.

Rule. Add together the coefficients, to their fum prefix the common fign, and fubjoin the common letter or letters.

CASE II. To add Terms that are like, but have unlike 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.

CASE III. To add Terms that are unlike.

Rule. Set them all down, one after another, with their figns and coefficients prefixed.

EXAMPLE.

2a+36

-50+8

2a+36-5c+8

Compound

Compound quantities are added together, by uniting the feveral Terms of which they confift, by the preceding rules.

The rule for Cafe III. may be confidered as the general rule for adding all Algebraical Quantities whatsoever, and, by the rules in the two preceding cafes, the like Terms in the quantities to be added may be united, fo as to render the expreffion of the fum more fimple.

PROB. II. To fubtract Quantities.

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

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

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

a

a+d-c, and is therefore the remainder

required.

PROB. III. To multiply Quantities.

General Rule for the Signs. When the Signs of the two terms to be multiplied arę like, the fign of the product is +; but, when the figns are unlike, the fign of the product is

CASE I. To multiply two Terms. Rule. Find the fign of the product by the general rule; after it place. the product of the numeral coefficients, and then fet down all the letters one after another, as in one word.

The reafon of this rule is derived from Def. 6. and from the nature of multiplication, which is a repeated addition of one of the quantities to be multiplied as often as there are units in the other. Hence alfo

the