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5. 73125 of £2 10s.
6. •4694 lb. troy. 7. £19 178. 31d. in pounds.
8. 141x285 in decimals. 9. Reduce 9d. to the decimal of a £. 10. Reduce 5 h. 48 min. 49.7 sec. to the decimal of a day. 11. What is •315 of 2 lbs. 7 oz. 15 dwt. troy ? 12. Multiply £3 4s. 6d. by 1.46875, and find the product
in £ s. d. 13. Divide £10 11s. 3d. by 29.25, and find the quotient
in £ s. d. 14. What common fractions are equal to 1:36 and •1634 ? 15. The time between one new or full moon and the next is
29.5305887 days: reduce the decimal to hours, minutes,
and seconds. 16. The circumference of a circle is 3.1416 times the
diameter : the earth's circumference is about 24857 miles : find its diameter, as near as can be depended on; the 6, in the foregoing decimal, being slightly too
great. 17. The diameter of the sun is about 883220 miles : find its
circumference, as near as can be trusted (see Ex. 16). 18. What is the value of •121875£ +178. 6 d. ? 19. What is the value of .875£+:37 crown? 20. Work the following by decimals: If 2 yrs. of sugar
cost £1 178. 6d., what will 1 cwt. 3 qrs. 21 lb. cost? 21. If 24 ac. 3 ro. 39 per. can be reaped in 12 hours, how
much can be reaped by the same hands in 15 h.
48 min. ? 22. If £6 138. be the wages of 8 men for 3.25 days, what
will be the wages of 20 men for 9.25 days? 23. Find the value of 34 of 26 of £2 138. ld. 24. Reduce ·47 of .23 of 78. 1 d. to the decimal of £1 14s. 8td.
(92.) Recurring, or Circulating Decimals. Before concluding the arithmetic of decimals, it is proper to say a few words about what are called recurring, or circulating decimals. They are so called, because the figures of which they consist continually recur, presenting either a constant repetition of the same figure, after a certain number of figures, or a repetition of the same set or row of figures: thus, •3333 ....is a recurring decimal ; so also is •7543543. and •592592.. &o. In the first of these instances, 3 is the recurring figure; in the second, 543 is the recurring period, as it is called ; and in the third, the recurring, or circulating period, is 592. Instead of repeating the period, it is customary to write it but once, and to put dots over the extreme figures of the period, by which we are to understand, that those figures recur without end: the three instances just noticed would thus be more briefly expressed as follows: 3, 7543, and 592. Decimals, of which the figures have this periodic character, very frequently present themselves in converting a common fraction into a decimal ; indeed, they always present themselves whenever the denominator of a vulgar fraction, in its lowest terms, is not entirely resolvable into factors consisting of 2's and 5's. You are aware, that, in order to convert a fraction into a decimal, we divide the numerator by the denominator, continually annexing zero after zero to the former, till the operation terminates of itself, or till we arbitrarily put a stop to it: and it is plain, that if we are at liberty to put as many O’s as we please to a number, that number, with the zeros attached to it, will become divisible, without remainder, by as many 2's and 5's as we please. Hence, a fraction whose denominator has no other simple factors but these, is always equal to a finite or terminable decimal : but, if other factors enter, or, which is the same thing, if, after removing all the 2's and 5's, a factor still exists in the denominator, then, the decimal, equal to the fraction, will be interminable, because this remaining factor, not being divisible by either 2 or 5, must terminate, either in a 1, a 3, a 7, or a 9; and no quotient-figure, multiplied into either of these, can ever produce a 0, as the terminating figure of the product, so that we might bring down O’s continually, without any hope of the work ending of itself: the following are a few instances. •1111.
·008497133 = .0909...=.09
= 4•7643 = •592592 ..
1 = 052631578947368421. As the last of these examples shows, a very simple fraction may give us a good deal of labour, before we can determine the circulating period of its equivalent decimal :* but, in a
* A method of abridging this labour was given by Mr. Colson, in Sir Isaac Newton's “ Fluxions.” An analogous method, much more convenient and expeditious, was proposed by the author of this Rudimentary Treatise, in Vol. xxxvi. of the Philosophical Magazine; the numbers for January and February, 1850.
1 11 16
case like this, such determination would be more curious than useful : it is easy to prove, however, that, the fraction being in its lowest terms, the period can never have so many figures as there are units in the denominator of the fraction : in the case of , for instance, we might be sure that the period could not have more than eighteen figures, wbich number we see it actually contains. The reason is this: the period can extend itself only so long as the successive remainders we arrive at, in carrying on the division, do not recur : whenever we come to a remainder, the same as one already employed, then, of course, the quotient-figures between the two must also recur; but this recurrence is postponed so long as the remainders continue to be all different; and as no remainder can be greater than the number which is a unit less than the divisor, it is plainly impossible that there can be more different remainders than is expressed by the divisor, minus 1. And this is, in general, all that we can say about the extent of the period, previously to actual trial.
Although recurring decimals are thus always interminable, you are not to infer, that interminable decimals are always recurring; those only are recurring which arise from the development, as it is called, of a vulgar fraction; such decimals are always convertible back again into the finite fractions to which they are equivalent; but many interminable decimal values occur in calculation which cannot be represented by a finite fraction, and which, therefore, are not recurring decimals. Of such decimals only, a finite portion of the interminable row can be used in computation, so that some amount of error in the abridged forms is unavoidable : in the preceding articles, I have shown how to exclude from any result that part of it which this imperfection would influence. When the decimals with which we work are recurring, the imperfection, consequent upon our using only a finite number of figures, can be rendered as minute as we please ; for one period being given us, we can always add on as many true decimals as we like, instead of employing zeros when we want additional places; to use zeros for such a purpose in recurring decimals, you will, of course, see, would be an intentional departure from strict accuracy: but, in dealing with circulating decimals, the way to avoid imperfection altogether, is to convert them into their equivalent vulgar fractions : the rule for this is as follows.
(93.) To convert a Recurring Decimal into its equivalent
Vulgar Fraction. RULE 1. Write down only the figures after the decimal point, up to the end of the first period, omitting the leading O's, if there be any, and consider the number thus written to be a whole number.
2. Subtract from this whatever portion of the decimal there may be which precedes the period, regarding this portion as a whole number also ; the remainder will be the numerator of the equivalent fraction.
3. For the denominator, write as many g's as there are figures or places in the period, followed by as many O's as there are decimals preceding the period. Prefix to the fraction whatever whole number may have been prefixed to the decimal.
Ex. 1. Convert :09 into its equivalent fraction.
Here, omitting the leading 0, we write 9 as a whole number; and as no decimals precede the period, the 9 will be the numerator of the required fraction, and 99 will be the denominator: therefore the fraction is a = i.
2. Convert 592 into its equivalent fraction. Here, agreeably to the rule, the fraction will be honey which, by dividing numerator and denominator by 37, reduces to 20
3. Convert 4.7543 into its equivalent fraction.
Here, the figures first to be written down are 7543, as a whole number, and from this, the whole number 7 is to be subtracted: the numerator is therefore 7536, and the denominator 9990; consequently, the fraction, with the whole number 4 prefixed, is 47580 = 41256 = 1875
Excercises. Reduce to fractions the following decimals. 135, 2.418, •5925, 00449, 3.7569, 621-621, 02439,
•857142, 1•0378, .008497133. To understand the principle of the rule, it will be sufficient to attend to the following simple and obvious cases, namely, 2 를 = '11111...
= .010101 ...., o o= :001001 . 9999 = .00010001..., &c., from which it appears, that a recurring decimal, whose period commences immediately after the decimal point, is converted into an equivalent fraction, thus: if the period consist of but one figure, this figure, taken
as a whole number, must be multiplied by }; if it consist of two figures, these, taken as a whole number, must be multiplied by Ág; if it consist of three figures, the number must be multiplied by ado; and so on : thus, 3 = or }; 1 = ; •592 = 56, &c. ; which is according to the rule. If the period do not commence immediately after the decimal point,
7.543 as, for instance, in 7643, then •7543 =
10 7 x 999 +543 7000-7+543 7543-7
9990 with the rule. In like manner, .27643 =
100 27 x 999 +543 27000—27+ 543 27543-27
99900 so on, agreeably to the rule.
5 4 3 999 10
5 4 3 9 9 9
(94.) INVOLUTION AND EVOLUTION. WHEN a set of equal factors are multiplied together, this particular case of multiplication is called involution, and the product is called a power of the number or factor, thus repeatedly used. If the number be simply multiplied by itself, the product is the second power, or square of that number: if the second power be also multiplied by the number, the product is the third power, or cube of that number: if, again, this be multiplied by the number, the product is the fourth power of that number: and so on, the number of times we thus use the same number as factor, always marking the power of that number, so that numbers may be raised to the fifth, sixth, seventh, or any other power, however great, provided only we have the patience to carry on these successive multiplications by it.
As this operation of involution is no more than common multiplication, there is nothing for me to explain in reference to the mode of performing it; and I here mention it, chiefly for the sake of showing you the meaning of a term in frequent use, and of introducing another particular in notation.
From what has just been said, you see that the second power, or square of any number, 3, for instance, is thus indicated: 3x3 = 9; that is, the square of 3 is 9 : that the