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5. Divide 478.325 by 1.43}, extending the quotient to 3 decimal places.
Ans. 332.942 +. 6. Divide 8972.436 by 756.3452, extending the quotient to 4 decimal places.
7. Divide 1 by 1.007633, extending the quotient to 6 decimal places.
Ans. .992425 +. 8. Find the quotient of .95372843 divided by 44.736546, true to 8 decimal places.
9. Reduce 433to a decimal of 4 places. Ans. .7448 +.
227. Common fractions can not always be exactly expressed in the decimal form; for in some instances the division will not be exact if continued indefinitely.
228. A Finite Decimal is a decimal which extends a limited number of places from the decimal point.
229. An Infinite Decimal is a decimal which extends an unlimited number of places from the decimal point.
230. A Circulating Decimal is an infinite decimal in which a figure or set of figures is continually repeated in the same order; as .3333+, or .437437437+.
231. A Repetend is the figure or set of figures continually repeated. When a repetend consists of a single figure, it is indicated by a point placed over it; when it consists of more than one figure, a point is placed over the first, and one over the last figure. Thus, the circulating decimals .55555+ and.324324324+, are written, 5 and .324.
232. A repetend is said to be expanded when its figures are continued in their proper order
number of places toward the right; thus, .24, expanded is .2424+, or .242424242+.
233. Similar Repetends are those which begin at the same decimal place or order; as 37 and .5, 2, and .375, 0.56 and 24.3.
234. Conterminous Repetends are those which end at the same decimal place or order ; as .75 and 1.53, .567, and 3.245.
Note.—Two or more repetends are Similar and Conterminous when they begin and end at the same decimal places or orders.
235. A Pure Circulating Decimal is one which contains no figures but the repetend; as .7, or .104.
236. A Mixed Circulating Decimal is one which contains other figures, called finite places, before the repetend; as .54, or .013245, in which .5 and .01 are the finite places.
PROPERTIES OF FINITE AND CIRCULATING DECIMALS. 237. The operations in circulating decimals depend upon the following properties.
Note.-1. The common fractions referred to are understood to be proper fractions, in their lowest terms.
I. Every fraction whose denominator contains no other prime factor than 2 or 5 will give rise to a finite decimal; and the number of decimal places will be equal to the greatest number of equal factors, 2 or 5, in the denominator.
For, in the reduction, every cipher annexed to the numerator multiplies it by 10, or introduces the two prime factors, 2 and 5, and also gives 1 decimal place in the result. Hence the division will be exact when the number of ciphers annexed, or the number of decimal places obtained, shall be equal to the greatest number of equal factors, 2 or 5, to be canceled from the denominater.
II. Every fraction whose denominator contains any other prime factor than 2 or 5, will give rise to an infinite decimal.
For, annexing ciphers to the numerator introduces no other prime factors than 2 and 5; hence the numerator will never contain all the prime factors of the denominator.
III. Every infinite decimal derived from a common fraction is also a circulating decimal; and the number of places in the repetend must be less than the number of units in the denominator of the common fraction.
For, in every division, the number of possible remainders is limited to the number of units in the divisor, less 1; thus, in dividing by 7, the only possible remainders are 1, 2, 3, 4, 5, and 6. Hence, in the reduction of a common fraction to a decimal, some of the remainders must repeat before the number of decimal places obtained equals the number of units in the denominator; and this will cause the intermediate quotient figures to repeat.
Notes.—2. It will be found that the number of places in the repetend is always equal to the denominator less 1, or to some factor of this number. Thus, the repetend arising from has 7—1= 6 places; the repetend arising from , has
- 5 places. 3. A perfect repetend is one which consists of as many places, less 1, as there are units in the denominator of the equivalent fraction.
4. If the denominator of a fraction contains neither of the factors 2 and 5, it will give rise to a pure repetend. But if a circulating decimal is derived from a fraction whose denominator contains either of the factors 2 or 5, it will contain as many finite places as the greatest number of equal factors 2 or 5 in the denominator.
IV. If to any number we annex as many ciphers as there are places in the number, or more, and divide the result by as many 9's as the number of ciphers annexed, both the quotient and remainder will be the same as the given number.
For, if we take any number of two places, as 74, and annex two ciphers, the result divided by 100 will be equal to 74; thus,
7400 - 100 74. Now subtracting 1 from the divisor, 100, will add as many units to the quotient, 74, as the new divisor, 99, is contained times in 74, (115, II); thus, 7400 • 99
or 7470; that is, if two ciphers be annexed to 74, and the result be divided by 99, both quotient and remainder will be 74. In like manner, annexing three ciphers to 74, and dividing by 999, we have
74000 – 999 : and the same is true of any number whatever.
V. Every pure circulating decimal is equal to a common fraction whose numerator is the repeating figure or figures, and whose denominator is as many 9's as there are places in the repetend.
For, if we take any fraction whose denominator is expressed by some number of 9’s, as , then according to the last property, annexing two ciphers to the numerator, and reducing to a decimal, we have
= 24; . In like manner, carrying the decimal two places farther, .24%= .2424; ; hence, o 24. By the same principle, we have ਨੂੰ =.2; ਹੇਨ .ji; =.02; 668 .óoi;=.324 ; and so on. And it is evident that all possible repetends can thus be derived from fractions whose numerators are the repeating figures, and whose denominators are as many 9's as there are repeating figures.
Note 5.—It follows from the last property, that any fraction from which a pure repetend can be derived is reducible to a form in which the denominator is some number of l’s; thus is
This is true of every fraction whose denominator terminates with 1, 3, 7, or 9.
VI. Any repetend may be reduced to another equivalent repetend, by expanding it, and moving either the second point, or both points, to the right; provided that in the result they be so placed as to include the same number of places as are contained in the given repetend, or some multiple of this number.
For, in every such reduction, the new repetend and the given repetend, when expanded indefinitely, will give results which are identical. Thus, .536 =.536536, or .536536536, or .5365, or .53653, or .5365365, or .53653653653; because each of these new repetends, when expanded, gives .53653653653653653653+.
Note 6.- If in any reduction, the new repetend should not contain the same number of places, or some multiple of the same number, as the given repetend, we should not have, in the expansions, the same figures repeated in the same order.
238. To reduce a pure circulating decimal to a common fraction. 1. Reduce .675 to a common fraction.
Analysis. Since the repetend has 3 places, we take for the denominator of
the required fraction the number expressed by three 9's, (237, V). Hence,
RULE. Omit the points and the decimal sign, and write the figures of the repetend for the numerator of a common fraction, and as many 9's as there are places in the repetend for the denominator.
EXAMPLES FOR PRACTICE.
1. Reduce .45 to a common fraction. 2. Reduce .66 to a common fraction. 3. Reduce .279 to a common fraction.
14 1 85
239. To reduce a mixed circulating decimal to a common fraction. 1. Reduce :0756 to a common fraction.
Analysis. Since .756 is equal .0756 = 9% = 16's
to 356, .0756 will be to of 750 2. Reduce .647 to a common fraction.
ANALYSIS. Reducing the finite .647 goo
part and the repetend separately 610 64
to fractions, we have +
To reduce these fractions to a 900 900
common denominator, we must 610 — 64 +7
multiply the terms of the first by 900
9; but the numerator, 64, may 647 .64
be multiplied by 9 by annexing
the first fraction reduced. The Or,
numerator of the sum of the two .647 given decimal.
fractions will therefore be 640 64 finite figures.
- 64 +7=583, and supplying 583
the common denominator, we have
In the second operation, 583 Ans.
the intermediate steps are omitted. 900'
Hence the following RULE. I. From the given circulating decimal subtract the finite part, and the remainder will be the required numerator.