Encyclopaedia Britannica, or, a Dictionary of arts, sciences, and miscellaneous literature : enlarged and improved. Illustrated with nearly six hundred engravings > Volume 1, A-AME

(
A L G E
Fractions. Mixed quantities may Ue multiplied after being re-
-v-—* duced to the form of fractions by prob. 3. Thus
1 bx\ a ob-A-bx a a^b-A-dbx ab-A-bx
b +-J x *= ^r~ x ;= -^r-= -v-
58. The reafon of the rule for multiplication may be
explained thus. If ^ is to be multiplied by c, the
jK^oduft will evidently be . but if it is only to be
multiplied by the former product muft be divided
by d, and it becomes 7^, which is the product required.
ba
Or let - — m. and-=://, then ct—bm and c—dn and
b d
a c oc
ac—bdmn : hence tnn or -r X Tj‘
b a ba
59. Prob. VIII. To Divide FraBions.
Rule. Multiply the denominator of the divifor by the
numerator of the dividend for the numerator of the
quotient. Then multiply the numerator of the di¬
vifor by the denominator of the dividend for the de¬
nominator of the quotient.
Or, multiply the dividend by the reciprocal of the di¬
vifor, the product will be the quotient required.
a c
Ex. 1. Divide by -.
r\a(ad . . . & d ad ,
the quotient required, or £ X - jrr as be"
fore.
_. a'1-\-ab 3a2
Ex. 2. Divide by 7.
2x a—b
a'2- ^ab* (a''—ah'2- a2 —b
3fl2 y+°b'(
a—b ) 2X \
the quotient.
n
m
ad'
BRA,
Sect. III. Of Involution and Evolution.
61. In treating of multiplication, we have obferved,
that when a quantity is multiplied by itfelf any num¬
ber of times, the product is called a power' of that
quantity, while the quantity itfelf, from which the
powers are formed, is called the root (j 36.) Thus a,
a2, and a1 are the firft, fecond, and third poivers of the
root a ; and in like manner
, and -7, denote the
n *
a a
fame powers of the root -•
62. But before confidering more particularly what
relates to powers and roots, it will be proper to obferve,
that the quantities A, A, _L? &;C. admit of being ex-
preffed under a different form j for, like as the quanti¬
ties a, a1, a1, &c. are expreffed a,spoJitive powers of the
root a, fo the quantities —
If either the divifor or dividend be an integer quan¬
tity, it may be reprefented as a fraftion, by placing unity
for a denominator ; or if it be a mixed quantity, it may
be reduced to a fraction by prob. 3. and the operation
of divifion performed agreeably to the rule.
60. The reafon of the rule for divifion may be ex-
C Cl
plained thus, let it be required to divide - by -. If
~ is to be divided by a, the quotient is —, but if it
is to be divided by then the laft quotient muft be
cb
multiplied by b ; thus we have — for the quotient re¬
quired. Or let ^ — and then azzbm and
bdn
c—dn; alfo adzzbdm and bczzbdn; therefore 7-7-=
bdm
be
, &.c. may be re-
„ a2 a5 1
fpectively expreffed thus, a~1, a-2, 3, &c. and con-
fidered as negative powers of the root a.
63. This method of expreffing the fradtions —, —7,
A, as powers of the root a, but with negative indices,
a*
is a confequence of the rule which has been given for
the divifion of powers j for we may confider — as the
quotient arifing from the divifion of any power of a by
the next higher power, for example from the divifion
I Cl2
of the 2d by the 3d, and fo we have — = —; butfince
powers of the fame quantity are divided by fubtradting
the exponent of the divifor from that of the dividend
A 2
(j 40.), it follows, that~ xza2~i=a'"z ; therefore the
fradtion — may alfo be expreffed thus, a~t. By confi-
a
dering — as equal to it will appear in the fame
manner that A == j and proceeding in this
way, we get A- A=a-J, A — A—&c. and fo
on, as far as we pleafe. It alio appears, that unity or
I may be reprefented by a0, where the exponent is a
a2
cypher, for i=—=zo*“1r:o0.
64. The rules which have been given for the mul¬
tiplication and divifion of powers with pofitive expo¬
nents will apply in every cafe, whether the exponents
be pofitive or negative j and this muft evidently take,
place, for the mode of notation, by which we reprefent
fradlional quantities as the powers of integers, but with
negative exponents, has been derived from thole rules.
Thus Axa3 or «“2 Xdl3:=0-1 + 3=«"“I=-, alfo—rX
a2 a x2
4 H 2 1