algebra. 
be taken must be prefixed ; thus 5 a denotes 
that the quantity a is to be taken 5 times, 
and 3 be represents three tim es be, and 7 
d 2 -J- 1 / denotes that \/ a 2 If is to 
be taken 7 times, &c. The numbers thus 
prefixed are called coefficients; and if a 
quantity have no co-efficient, unit is under- 
stood, and it is to be taken only once. Si- 
milar or like quantities are those that me ex- 
pressed by the same letters under the same 
powers, or which differ only in their co-effi- 
cients; thus, 3 be, 5 be, and 8 6 c, are 
like quantities, and s o are the radicals 
2 * / iil and 7 S-if. But unlike 
v u « 
quantities are those which are expressed 
by different letters, or by the same let- 
ters with different powers, as 2 a b, 5 a b 2 , 
and 3 a 2 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 call- 
ed a simple quantity. But the quantity 
which consists of two or more such simple 
quantities, connected by the signs or — , 
is called a compound quantity ; thus, a — 
2 a 6 -{- 5 a 6 cisa compound quantity; and 
the simple quantities a, 2 a 6 , babe, are 
called its terms or members. If a com- 
pound 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 quan- 
tity is expressed may be ranged at plea- 
sure, and yet retain the same signification ; 
thus a b and 6 a are the same quantity, the 
product of a and 6 being the same with that 
of 6 by a. The several terms of which any 
compound quantity consists may be dis- 
posed in any order at pleasure, provided 
they retain their proper signs. Thus, a — 
2 ab + 5a 2 b may be written a -j- 5 a 1 b — 
•lab, or — 2 a b -J- a -f- 5 a 2 b, for all these 
represent the same thing or the quantity 
which remains, when from the sum of a and 
5 a 1 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, th* remainders will be equal. 
3. If equal quantities be multiplied by the 
same, or equal quantities, the products will 
be equal. 
4. If equal quantities be divided by the 
same, or equal quantities, the quotients will 
be equal. 
5. If the same quantity be added to and 
subtracted from another, the value of the 
latter will not be altered. 
6 . If a quantity be both multiplied and 
divided by another, its value will not be al- 
tered. 
ADDITION OF ALGEBRAICAL QUANTITIES. 
The addition of algebraical quantities is per- 
formed by connecting those that are unlike with 
their proper signs, and collecting those that are 
similar into one sum. 
Add together the following unlike quanti- 
ties : 
Ex. 1. ax 
— bu , 
+ 3* 
— 2 y 
Ans. a x — bu-\- 3% — 2 y 
Ex. 2. — a -f- b 
— }— 3 » — x 
— 4 y -}- 3 c 
Ans. — a +6 + 3c — x — 4 ^ — (— 3 s 
It is immaterial in what order the quanti- 
ties are set down, if we take care to prefix 
to each its proper sign. 
When any terms are similar, they may be 
incorporated, and the general expression for 
the sum shortened. 
1. When similar quantities have the same 
sign, .their sum is found by taking the sum 
of the co-efficients with that sign, and an- 
nexing the common letters. 
Ex. 3. 4 a — 5 b 
2 a — 6 6 
9a — 36 
Ans. 15 a — 14 6 
Ex. 4. 4 a 2 c — tOJtfe 
6 a 2 c — 9 b de 
11 a 2 c — 3b de 
Ans. 21 a 2 c — 22 6 de 
The reason is evident ; 4 a to be added, 
together with 2 a 'and 9 a to be added, makes 
1 5 a to be added ; and 5 J to be. subtracted, 
