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which it will be converted into a decimal, is the multiplication of the numerator by some power of 10, the division of the product by the original denominator, and the pointing off as decimals as many figures in the quotient, as ciphers have been added to the numerator. Now in order that the decimal may be finite, the multiplied numerator must be divisible by the original denominator; therefore the factors of this denominator must be factors of the numerator; but the only factors of the numerator are those of the original numerator, (and by hypoth. none of these are common to the denominator,) and 2's and 5's, which are the only factors of powers of 10. Hence the factors of the denominator must be only 2’s and 5's in order that the decimal inay be finite. And moreover, there must be in the numerator as many 2's and 5's as in the denominator; hence since 2's and 5's enter in equal numbers into any power of 10, the power of 10, by which the numerator must be multiplied, is the highest power, whether of 2's or 5's in the denominator ; so that if a vulgar fraction be convertible into a finite decimal, there will be a number of decimal figures equal to the number of 2's or 5's in the denominator, whichever is the greater.
In all cases in which the denominator of a fraction in its lowest terms has any other divisors than 2's and 5's, since its factors cannot, by multiplication by any power of 10, be made to enter into the nnmerator, it cannot be converted into a finite decimal. In these cases, it will appear that the figures recur in a fixed order, and that the extent of a recurring period is always less than the denominator. For in dividing the decimals by the denominator, every remainder must be less than the divisor, therefore at the most there can only occur a number of different remainders less by one than the denominator, and consequently within a number of divisions equal to the denominator some remainder must necessarily occur, which has occurred before; therefore, in the next division, there will be the same dividend as has occurred before, therefore the same quotient, therefore the same remainder, and so on; so that it is evident that the decimal figures will recur in the same order as before, and the greatest number of figures in a recurring period is less by one than the denominator.
The same Prop. exhibited algebraically .- Let be the fraction in its lowest terms; then 등
Now in order that may be con
b 6 X 10P vertible into a finite decimal, a X 10P must be divisible by b; hence the factors of b must be factors of a X 10P; but none of the factors of b are among those of a, therefore if they are factors of a X 10P they must be among those of 10p ; now the only factors of 10P are 2's and 5's, therefore the factors of b also must be only 2's and 5's. Let then b = 2 X 5s: then
a X 10p
that is if
be converted into a finite decimal, it will have r or s 27 x 58 decimal places, according as 70 > or Ć s.
If b be of any other form than 27 X 5s its factors cannot be among those of a X 10p, therefore, will not be convertible into a finite decimal. In this case it is evident that, in dividing, every remainder must be less than b; therefore at the most there can only be 6-1 different remainders : hence within b operations upon the decimal, some remainder must occur, which has occurred before; therefore in the next division there will be the same dividend, therefore the same quotient, the same remainder, and so on; so that it is evident that the decimal figures will recur in a fixed order, and that the greatest possible number of recurring figures will be b-1.
Prop. 44.—To prove the Rule for the conversion of a re
curring decimal into a vulgar fraction.
99 999 and so on, the number of figures in the recurring period being the number of nines in the denominator, and since every pure recurring decimal is the product of one of these, and the number composed of the recurring figures, the rule for conversion of a pure recurring decimal into a vulgar fraction is evident.
3635 Thus .3635 = 3635 x .0001 = 3635 x
9999 9999 If the decimal be a mixed circulator, let it be multiplied by such a power of 10, as will bring the non-recurring figures on the left of the decimal point; the value of the decimal may now be found, and that of the original decimal by dividing by the power of 10, by which it was multiplied. Thus suppose it be required to find the vulgar fraction equivalent to .5634) ; this multiplied by 100 becomes 56.341, which is equivalent to the vulgar
341 56 X (1000 — 1) + 341 56341 - 56 fraction 56—- or
; hence the deci999 999
56341 - 56 mal .56341 is equivalent to
i.e. to a fraction, whose numerator
99900 is the difference between all the figures of the decimal and the non-recurring; and whose denominator is composed of a number of nines equal to the re
curring, followed by a number of ciphers equal to the non-recurring, figures. The same may be shewn of every similar decimal.
The same Prop. exhibited algebraically:-Let .P Q represent a recurring decimal, P denoting the p non-recurring figures, Q the q recurring;
let x = .PD
10P+q x = PQ.
10P(109 — 1) Now PQ represent all the figures of the decimal, P the non-recurring ; and 109 1 will consist of g nines, and the multiplication of this by 10P will introduce on the right of these p ciphers. Whence the Rule.
Prop. 45.—To shew that the ratio of one number or quan
tity to another may be properly represented by the fraction, whose numerator is the number of units in the former, and denominator the number of the same kind
of units in the latter, quantity. Ratio is the relation which one quantity bears to another of the same kind, this relation being determined by considering what multiple, part, or parts, the one is of the other. Now if the quantities be expressed in terms of the same unit, the unit may be expressed as a multiple, part, or parts, of the one quantity, and hence the other quantity may be expressed as a multiple, part, or parts of this. Thus suppose that it be required determine what multiple, part, or parts, the length 2 feet is of the length 3 yards, if we express these lengths in terms of the same unit either a foot, or a yard, they will be 2 feet and 9 feet; or 3 yard, and 3 yards: hence it
appears that the unit 1 foot is š of 9 feet or three yards; and the unit 1 yard is } of 3 yards; and therefore the length 2 feet is g of 3 yards; or the length § yard is į of į of 3 yards'; or š x } of 3 yards, or of 3 yards. Hence g or (which are equivalent) represent the part which 2 feet are of
3 3 yards: and the numerators and denominators of these factors are respectively the number of units in the two lengths.
Again, suppose it required to find the ratio of the abstract number 2 to the abstract number 9; that is to find “how often” 9 is contained in 2. The meaning of this is, (Prop. 35,) that we have to find what operation must be performed upon the result of that denoted by 9 to produce the result of that denoted by 2. Now we know that 2=9X ; therefore is the symbol of the operation required, or expresses the ratio of 2 to 9.
Cor. 1. Hence the ratio of one number or quantity to another may be obtained by dividing the number of units in the former by the number of like units in the latter.
Cor. 2. Hence the terms of a ratio may be simplified by multiplying or dividing both by the same number; since we may do this to the terms of the equivalent fraction without altering its significance.
Cor. 3. Hence also two ratios may be compared by comparing the equivalent fractions, in order to which they must be reduced to a C. D. and as any C. D. will serve the purpose, we may use that which is formed by the product of the two: so that the comparison will be made by comparing the products of the antecedent of the one by the consequent of the other, the antecedents and the consequents of the ratios being the numerators and denominators of the fractions.
Prop. 46.--To shew how to divide a number or quantity
into parts, which shall bear to each other a given ratio. If the number or quantity be divided into equal parts, the number of parts being the sum of the numbers composing the ratios, and if collections of these parts be formed, containing numbers eqnal to the several numbers in the ratios, it is evident that the parts so formed will be in the required ratios. Thus suppose it required to divide 20 into 3 parts in the ratio of 4, 5, and 9. Dividing 20 into 18 parts, each is 10; and taking 4, 5, and 9 of these parts, the numbers *°, 5., 9 are formed, which are in the ratio of 4, 5, and 9; for * ; and = 0 = g.
The same Prop. exhibited algebraically :--Let it be required to divide the number N into erts in the ratio of a, b, c: let x, y, z be the parts: then y b
y -=-:or-=-=-=t suppose: hence x = at;y=bt; b
N z=ct; and x +y+z=N=(a+b+c)t: whence t =
Prop. 47.-To shew that if four numbers be proportional
in a given order, the product of the extremes is equal to
that of the means, and conversely. Four numbers are proportionals when the ratio of the first to the second is equal to that of the third to the fourth. Hence these ratios must be compared, which is effected by comparing the products of the antecedent of the one by the consequent of the other. If these products be equal, the ratios are equal, and the numbers are proportional. But these pro
ducts are respectively the products of the extremes, and of the means, of the four numbers. Hence the truth of the Prop.
The same Prop. exhibited algebraically:-Let a, b, c, d, be four numbers ; if they be proportional in this order, then 7 = or (reducing to a
d ad C. D.) bd bd
whence a d=bc; or the product of the extremes is equal to that of the means. Conversely:-if four numbers be such that the product of one pair is equal to the product of the other pair, they are proportionals, the numbers in the products being either both extremes, or both means. For if a Xd=bXc, then, dividing by 6 X d,
=-, i.e. b
b d a :b:: c:d; or, dividing by c X d, 6
i.e. a :C:: b: d; or d b
d again, dividing by a Xc, -=-, i.e. b: a :: d:c; or again, dividing by
d а Хь,
i.e. c:a:: d: 6. b Cor. 1. Hence, if four numbers are proportional, the second is to the first as the fourth to the third. For if a : 6 :: c:d, a xd=bXc; .. from the Prop. b : a :: d:c.
Cor. 2. Also, in the same case, the first is to the third as the second to the fourth. For if a : 6 :: c:d, a Xd=b Xc; and .. from the Prop. a: 0 :: 6: d.
Cor. 3. Also the third is to the first as the fourth to the second. For by Prop. c :a:: d:b.
Cor. 4. Hence also either mean and extreme may be multiplied or divided by the same number without affecting the proportion, for m a Xnd =mb Xnc. Prop. 48.- To find a fourth proportional to three given
numbers. Since the product of the second and third of four numbers, which are proportionals, is equal to that of the fourth and first, therefore the former product divided by the first is equal to the fourth, which is the number required.
ъхс Thus if a, b, c, d, be the four numbers, a Xd=bXc, ..d =
Prop. 49.—To find a third proportional to two given
numbers. If three numbers are continued proportionals, the first is to the second, as the second to the third, therefore the product of the first and third is equal to the square of the second. Hence the third is obtained by dividing the square of the second by the first. Thus if a, b, c, be the four numbers,
12 a : b :: 6:2;
.. a Xc=6%, and c=