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

646
ALGEBRA.
Equations raoft be divifible by P, and hence the truth of the pro-
with ratio- pofition is manifeil.
rial Roots.^ 241. It follows from this propofition, that if the
prime number P, which we have fuppofed not to be a
divifor of B, is at the fame time not a divifor of A, it
cannot be a divifor of AB the produdl of A and B.
242. Let — be a fraftion in its loweft terms, then
a
the numbers a and b have no common divifor ; but
from what has been juft now (hewn, it appears, that if
a prime number be not a divifor of a it cannot be a
divifor of ay, a or a', and in like manner, that if a
prime number is not a divifor of b, it cannot be a di¬
vifor of byb, ov b1; therefore, it is evident that n*
and b1 have no common divifor, and thus the fradtion
b*
-r is alfo in its loweft terms.
a
Hence it follows that the fquare of any fradtional
quantity is ftill a fradion, and cannot pofixbly be a
whole number ; and, on the contrary, that the fquare
root of a whole number cannot poffibly be a fradion j
fo that all inch whole numbers as are not perfed
fquares can neither have their roots exprefl’ed by inte¬
gers nor by fradions.
243. Since that if a prime number is not a divifor
of 0, it is alio not a divifor of a*, therefore if it is not a
divifor of 0, it cannot be a divifor oi ay a1 or a3, § 241,
and by reafoning in this way, it is obvious that if a
prime number is not a divifor of 0, it cannot be a di¬
vifor of an ; alfo, that if it is not a divifor of Z>, it can¬
not be a divifor of bn, therefore if — is a fradion in its
a
loweft terms, —- is alfo a fradion in its loweft terms ;
fo that any power whatever of a fradion is alfo a frac¬
tion, and on the contrary, any root of a whole num¬
ber is alfo a whole number. Hence it follows that if
the root of a whole number is not expreflible by an in¬
teger, fuch root cannot be exprefled by a fradion, but
is therefore irrational or incommenfurable.
244. Let us next fuppofe that
+ f-T*+U=°
is any equation W'hatever, in which P, Q, &c. denote
integer numbers •, then if its roots are not integers they
cannot poftibly be rational fradions. For, if poffible,
let us fuppofe x— -, a fradion reduced to its loweft
' a
terms, then, by fubftitution,
0” 0’“"* ^ 0,,—1 0
T»+ • • * +l£+U=o,
and, reducing all the terms to a common denominator,
0"+P0n-l£ -FQ0"-*£ f T0^‘-1+U^=o,
which equation may alio be expi-effed thus,
0n-f-^(P0"-1 +Q0n“*£ |-T«£"_2+U bn~r)=0,
■where the equation confifts of two parts, one of which
is divilible by b. But by hypothefis a and b have no
common meafure, therefore 0” is not divifible by
§ 243 *, hence it is evident that the two parts of the
equation cannot deftroy each other as they ought to do}
therefore x cannot poffibly be a fradion.
3
o vttt _, /• 7 . , Approxi-
oect. XVI. Refolution of Equations by Approxitna- mation.
tion. —v-—
245. When the roots of an equation cannot be ac¬
curately exprefled by rational numbers, it is necelfary
to have recourfe to the methods of approximation, and
by thefe we can always determine the numerical values
of the roots to as great a degree of accuracy as wre
pleafe.
246. The application of the methods of approxima¬
tion is rendered eafy by means of the following princi¬
ples :
If two numbers, either whole or fradional, be found,
which, when fubftituted for the unknown! quantity in
any equation, produce refults with contrary figns } we
may conclude that at leaf! one root of the propofed
equation is between thofe numbers, and is confequentlv
real.
Let the propofed equation be
a?3—yx1 -|-1 ox—15=0,
which, by colleding the pofitive terms into one fum,
and the negative into another, may alfo be expreffed
thus,
at3-|-io.v—(5#*-}-i5)=o}
then to determine a root of the equation, we muft find
fuch a number as when fubftituted for x will render
a;3-|- ioat:=5#*+ 15.
Let us fuppofe x to have every degree of magnitude
from o upwards in the fcale of number, then &3-{-io#
and 5#*-ft 15 wrill both continually increafe, but with
different degrees of quicknefs, as appears from the fol¬
lowing table.
Succeffive values of x. o, 1, 2, 3, 4, 5, 6, &c.
—— ■ of AT3-flOW. 0,11, 28, 57, IO4, I75, 276, See.
of 5**-f IJ. 15,20, 35, 60, 95,140, 195, &c.
By infpefting this table, it appears that while x in-
creafes from o to a certain numerical value, which ex¬
ceeds 3, the pofitive part of the equation, or a:3-ftlow,
is always lefs than the negative part, or $x* -ft 15 } fo
that the expreffion
w3-ft iow—(5#*+15), or w3—5«*-ftiow—15
muft neceffarily be negative.
It alfo appears that when w has increafed beyond that
numerical value, and which is evidently lefs than 4,
the pofitive part of the equation, inftead of being lefs
than the negative part, is now greater, and therefore
the expreffion
w3—jw* + low—15
is changed from a negative to a pofitive quantity.
247. Hence we may conclude that there is fome
real and determinate value of w, which is greater than
3, but lefs than 4, and which wdll render the pofitive
and negative parts of the equation equal to one another}
therefore that value of w muft be a root of the propofed
equation } and as what has been juft now Ihewn in a
particular cafe will readily apply to any equation what¬
ever, the truth of what has been afferted at § 246 is
obvious.
248. Two