A L G E B It A. 
51 
the quotient. Thus, the quotient in the last ex- 
2x- 
ampie is found to be a — x -J . And 
a x 
lb ah, divided by b — a, gives for the quo- 
, . ?ab 
tient b -f- . 
b — a 
Not e. The sign —- placed between any two 
quantities, expresses the quotient of the former 
divided by the latter. Thus a - r b a — x 
is the quotient of a -\-b divided by a — x. 
Of Fractions. 
In the preceding page it was said, that the 
quotient of any quantity a divided by b, is ex- 
pressed by placing a above a small line, and b 
under it, thus, 
These quotients are also 
called fractions ; and the dividend or quantity 
placed above the line is called the numerator 
of the fraction, and the divisor or quantity 
placed under the line i3 called the denominator. 
<2 
Thus, expresses the quotient of 2 divided 
bv 3 ; and 2 is the numerator and 3 the deno- 
minator of the fraction. 
If the numerator of a fraction is equal to the 
denominator, then the fraction is equal to unity. 
a b 
Thus, — , and —— , are equal to unit. If the 
a b 
numerator is greater than the denominator, 
then the fraction is greater than unit. In both 
these cases, the fraction is called improper. 
But if the numerator is less than the denomi- 
nator, then the fraction is less than unit, and 
5 , 
is called- proper. Thus — is an improper 
fraction ; but — and — are proper fractions. 
A rnixt quantity is that whereof one part is an 
4 
integer, and the other a fraction ; as 3 — ■, and 
2 1 l_ 
a — ; and a -j — — . 
3 1 b 
Pros. I. To reduce a Mixi Quantity to an Impro- 
per Fraction. 
Rule. Multiply the part that is an integer 
by the denominator of the fractional part ; and 
to the product add the numerator ; under their 
sum place the former denominator. 
Thus 22, reduced to an improper fraction, 
a + T = — . 
and a — x -J- 
Prob. II. To reduce an Improper Fraction to a Mixt 
Quantity. 
Rule. Divide the numerator of the fraction 
by the denominator, and the quotient shall give 
the integral part ; the remainder set over the de- 
nominator shall be the fractional part. 
12 2 ah 4- a 2 a 2 
I has, — - = 2 — ; J — - xx a - j- - — — ; 
la 4- xx _ , 
— a -j- 
a — x 
Trob. III. To reduce Fractions of different Denomi- 
nators, to Fractions of equal Value that shall have 
• the same Denominator. 
Rule. Multiply each numerator, separately 
taken, into all the denominators but its own, and 
the products shell give the new numerators. 
Then multiply all the denominators into one an- 
other, and the product shall give the common 
denominator. 'Tints, 
The fractions 
equal to these fractions, 
— , are respectively —r- — n, and it wdl be a ~ mb, c xx nb, and 
d °* 
acd bbd ccb , . , , , , , . , a "I - c . ,.i „ „ 
a -j- c, ana m -j- n xz. — - — , that 
— which mb -4- nb 
bed y bed ’ bed 
have the same denominator bed. And the frac- 
tions 2., 2 ? A, are respectively equal to these 
±2. A 5 ±3 
6 O’ 6 O’ 6 O' 
Prob. IV. To Add and Subtract Fractions. 
Rule. Reduce them to a common denomina- 
tor, and add or subtract the numerators; the 
sum or difference set over the common denomi- 
nator, is the sum or remainder required. 
d ade -j- bee -]- d 2 b 
d . e bde 
ad — be 2 .3 
a , c a 4- c 
— — . After the same manner, 
b ~ b b 
3. Isay, r X— .(= 
X «) 
— ; for 
hi 
Thus, — -f 
8-[-9 
bd 
3 
12 
12 
16 — IS 
20 
3.v 
2 3 6 6 
Prob. V. To Midtiply Fractions. 
Rule. Multiply their numerators one into 
another to obtain the numerator of the product; 
and their denominators multiplied into one an- 
other, shall give the denominator of the product. 
r „, a c ac 2 4 8 
1 bus, 
and 
If a mixt quantity is to be multiplied, first re- 
duce it to the form of a fraction (by Prob. I.). 
And if an integer is to be multiplied by a frac- 
tion, you may reduce it to the form of a frac- 
tion by placing unit under it. 
5 3- X ^T 
9 2 18 , , bx 
X — — = 6; b A 
' ' o a ’ 1 
1 " 3 
ba -j- bx 
b + 
ab 
a 2 b -j- abx ab b e 
ax JC 
Prob. VI. To Divide Fractions. 
Rule. Multiply the numerator of the divi- 
dend by the denominator of the divisor, their 
product shall give the numerator of the quoti- 
ent. Then multiply the denominator of the di- 
vidend by the numerator of the divisor, and 
their product shall give the denominator. Thus 
\ 2 1 
f 10 
3x5/ 
f 35 
e \ a ( 
J 3 ' 
c 12’ 
7 j 8 V 
i 24’ 
d) b [ 
a 4“ b \ a ■ 
— b / a 2 — 
2 ab -f- b 2 . 
a — b' a ' a 1 -j- a b 
These last four Rules are easily demonstrated 
from the definition of a fraction. 
1. It is obvious that the fractions . — — , 
b ’ d ’ 
e -ii a df c kf e bd 
/> “* equal f ^ 
since, if you divide adfby bdf the quotient will 
be the same as of a divided by b ; and df di- 
vided by dbf, gives the same quotient as c di- 
vided by d\ and ebd divided by fbd, the same 
quotient as s divided by f 
2. Fractions reduced to the same denomina- 
tor are added by adding their numerators and 
subscribing the common denominator. I say 
c a c a 
- T — ■ — Z — ' For > cal1 T - = tn, and 
9 9 V 
C 2 
bm — a, dn xx c ; and bdmn xx ac, and mn xx 
ac . a c ac 
; that is, -rrr X — = 
bd b d bd 
a c m , ad 
4. I say; — - divided by - — , or — , gives — — ; 
b d n co 
for mb xx a, and mbdxx ad ; nd xx c, and ubd xx cb ; 
mbd ad . m ad 
therefore — - — — ; that is, — = 
nbd cb n cb 
Prob. VII. To find the greatest common Measure of 
t-nvo Numbers ; that is, the greatest Number that 
can divide them both nuiihout a Remainder. 
Rule. First divide the greater number by the 
lesser, and if there is no remainder, the lesser 
number is the greatest common divisor required. 
If there is a remainder, divide your last divisor 
by it ; and thus proceed continually dividing the 
last divisor by its remainder, till there is no re- 
piainder left, and then the last divisor is the 
greatest common measure required. Thus, the 
greatest common measure of 45 and 63 is 9; and 
the greatest common measure of 256 and 48 is 16. 
45) 63 (1 48) 256(5 
45 . 240 
18) 45 (2 
36 
16) 48 (3 
48 
9) 18 (2 
18 
Much after the same manner the greatest 
common measure of Algebraic quantities is dis- 
covered ; only the remainders that arise in the 
operation are to be divided by their simple divi- 
sors, and the quantities are always to be ranged 
according to the dimensions of the same letter. 
Thus, to find the greatest common measure of 
a 2 — b 2 and a 2 — 2 ab -{- b 2 x, 
a 2 — b 2 ) a 2 - 2 ab 4- b 2 (1 
a 1 — b 2 
— 2 ab 4 - 2 b 2 remainder, 
which divided by — 2b is reduced to 
a — b) a 2 — b 2 ( a b 
a 2 - b 2 
O 0 
Therefore a — 6 is the greatest common mea- 
sure required. 
The ground of this operation is, that any 
quantity that measures the divisor and the re- 
mainder (if there is any) must also measure the 
dividend ; because the dividend is equal to the 
sum of the divisor multiplied into the quotient, 
and of the remainder added together. Thus, ia 
the last example, a — l> measures the divisor 
a 2 — b 2 , and the remainder — 2ab-\-2b\ it must 
therefore likewise measure their sum a 2 — 2al 
4 -?■ You must observe in this operation, to 
j make that the dividend which has the highest 
powers of the letter according to which the 
quantities are ranged. 
Prob. VIIT. To reduce any Fraction to its hzi’at 
Terms. 
Rule. Find the greatest common measure of 
the numerator and denominator; divide them 
by that common measure, and place the quo- 
