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be taken must be prefixed; thus 5 a denotes that the quantity a is to be taken 5 times, and 3 b c represents three times bc, and 7
A/F-F7 denotes that Vo-F to is to be taken 7 times, &c. The numbers thus prefixed are called co-efficients; and if a quantity have no co-efficient, unit is under. stood, and it is to be taken only once. Similar or like quantities are those that are expressed by the same letters under the same powers, or which differ only in their co-efficients; thus, 3 b c, 5 b c, and 8 b c, are like quantities, and so are the radicals 2 v/ *H: and 7 v/ “H. But unlike quantities are those which are expressed
by different letters, or by the same letters with different powers, as 2 a b, 5 a b”, and 3 a” b. When a quantity is expressed by a single letter, or by several single letters multiplied together, without any intervening sign, as a, or 2 a b, it is called a simple quantity. But the quantity which consists of two or more such simple quantities, connected by the signs-H or—, is called a compound quantity; thus, a 2 a b-H 5 a b c is a compound quantity; and the simple quantities a, 2 a b, 5 a b c, are called its terms or members. If a compound quantity consist of two terms, it is called a binomial; of three terms, a trinomial; of four terms, a quadrinomial, &c. of many terms, a multinomial. If one of the terms of a binomial be negative, the quantity is called a residual quantity. The reciprocal of any quantity is that quantity inverted,
or unity divided by it; thus ; is the reciprocal of }, and; is the reciprocal of a. The letters by which any simple quantity is expressed may be ranged at pleasure, and yet retain the same signification; thus ab and ba are the same quantity, the product of a and b being the same with that of b by a. The several terms of which any compound quantity consists may be disposed in any order at pleasure, provided they retain their proper signs. Thus, a2 a b +5 a'b may be written a +5 a b – 2 a b, or –2 a b-Ha-H 5 a'b, for all these represent the same thing or the quantity which remains, when from the sum of a and 5 a b, the quantity 2 a b is deducted.
Axioms. 1. If equal quantities be added to equal quantities, the sums will be equal.
2. If equal quantities be taken from equal quantities, the remainders will be equal,
Ex. 6. 6 a + 4 b + 9 c
—9 a + 3 b + 16 c
+ 12 a - 7 b – 20 c
Ans. 9 a * + 5 c In the first part of the operation we have 7 times a to add, and 5 times a to take away; therefore, upon the whole, we have 2 a to add. In the latter part, we have 3 times b to add, and 9 times h to take away; i. e. we have, upon the whole, 6 times b to take away: and thus the sum of all the
quantities is 2 a-6 b.
If several similar quantities are to be added together, some with positive and some with negative signs, take the difference between the sum of the positive and the sum of the negative co-efficients, prefix the sign of the greater sum, and annex the common letters. Ex. 7. 3 a” +4 b c – e’ + 10 r—25 —5 a” + 6 b c + 2 e” — 15 a +44 — 4 a”—9 b c – 10 e” +21 ar—90
sign of the quantity to be subtracted, and then adding it to the other by the rules laid down in the last article. o Ex. 1. From 2 bar take cy, and the difference is properly represented by 2 bar—cy; because the – prefixed to c y shews that it is to be subtracted from the other; and 2 b r— cy is the sum of 2 bar and — cy. Ex. 2. Again, from 2 b ar take — cy, and the difference is 2 r + cy; because 2 br= 2b r + c y—cy, take away—cy from these equal quantities, and the differences will be equal; i. e. the difference between 2 b r and – c y is 2 b ar–H cy, the quantity which arises from adding + c y to 2 br.
give a b; which we know is b. From this consideration are derived all the rules for the division of algebraical quantities. If the divisor and dividend be affected with like signs, the sign of the quotient is +: but if their signs be unlike, the sign of the quotient is —. If– a b be divided by —a, the quotient is + b; because — a x + b gives — a bi and a similar proof may be given in the other cases. In the division of simple quantities, if the co-efficient and literal product of the divisor be found in the dividend, the other part of the dividend, with the sign determined by the last rule, is the quotient. Thus, #=e; because a b multiplied by • gives a b c. If we first divide by a, and then by b, the
The reason of this, and the foregoing rule, is, that as the whole dividend is made up of all its parts, the divisor is contained in the whole, as often as it is contained in all the parts. In the preceding operation we inquire first, how often a is contained in a’, which gives a for the first term of the quotient, then multiplying the whole divisor by it, we have a” — a b to be subtracted from the dividend, and the remainder is — a b-H bo, with which we are to proceed as before.
The whole quantity a' — 2 a b-H bo is in reality divided into two parts by the process, each of which is divided by a-b; therefore the true quotient is obtained.
The quantity 2 r", found in every term of one of the divisors, 2 a” r" — 2 r", but not in every term of the dividend, a”—a’ x — a x* +r', must be left out; otherwise the quotient will be fractional, which is contrary to the supposition made in the proof of the rule; and by omitting this part, 2 at, no common measure of the divisor and dividend is left out; because, by the supposition, no part of 2 x' is found in all the terms of the dividend.
To find the greatest common measure of three quantities a b c ; take d the greatest common measure of a and b, and the greatest measure of d and c is the greatest common measure required. In the same manner, the greatest common measure of four or more quantities may be found.
If one number be divided by another, and the preceding divisor by the remainder, according to what has been said, the remainder will at length be less than any quantity that can be assigned.