number of decimal places that are to be obtained. For most purposes it will be sufficient, unless it is otherwise desired, to stop the division at the fourth or fifth decimal place.

(4.) Reduce 15 to a decimal of five places.

7)15.00000

2.14285 Answer.

decimal of the required The principle on which

or, as 1524, the fraction may be turned into a number of places, and the integer 2 then prefixed to it. the rule is founded is simply this: when ciphers are added to the numerator of the vulgar fraction, an equal number of ciphers are supposed to be added also to the denominator; both terms are then divided by the original denominatorthe fraction sustaining no change of value by either of these operations—and from the result, a decimal is formed by pointing off as many places as there are ciphers in the denominator. Thus,

with a cipher annexed to numerator and denominator becomes 10, which divided in both terms by 2 (the original denominator)

= 10 = 5.

10000

70000 160000'

with 4 ciphers annexed to numerator and denominator becomes and this divided in both terms by 16 (the original denominator) = 4375 = .4375. Very few vulgar fractions are convertible into exactly equivalent decimals. If the denominator of the vulgar fraction, when that fraction is reduced to its lowest terms, consist of any other number than 2 or 5, or the product of some number of 2's and 5's multiplied together, the corresponding decimal must be interminate the process of division will invariably leave a remainder, since 2 and 5, or their products, when multiplied by themselves, or by one another, are the only measures of 10, 100, 1000, &c.

For instance, is convertible into the terminate decimal 25, because the denominator 4 is the product of 2 x 2, and 2 is a measure of 10. Again, may be exactly decimalised, since 16 is the product of 2×2×2×2. But such fractions as,,,have no exact decimal equivalents, as none of the denominators can be formed by multiplying together a certain number of 2's or 5's, and will not, therefore, divide without remainder, such numbers as 10, 100, 1000, &c.

It must be evident that the formation of a terminate or an interminate decimal from a vulgar fraction depends entirely on the character of the denominator, let the numerator be what it may. Unless the denominator be a measure of 10, 100, &c., it cannot be a measure of 10, 100, &c., multiplied by any other number.

Take as examples, and 8, is 3 × 1, and as is reducible to the terminating decimal 2, so is reducible to a terminating decimal three times as great, or 6. In other words, since 5 is a measure of 10, it must also be a measure of three times 10. Again, 8×, and as 11 is not a measure of 100, 1000, &c., neither is it a measure of 8 times 100, 8 times 1000, &c. The decimal, therefore, which corresponds to is necessarily interminate.

Although but few vulgar fractions have precise decimal equivalents, the usefulness of the decimal system is not, on that account, in the smallest degree diminished. Nicety in the expression of fractional quantities is never required beyond a certain assignable point. The most delicate instrument of measurement that has yet been devised will hardly enable us to appreciate, as a sensible magnitude, the tenmillionth part of an inch. The finest balance used in scientific experiments is

barely affected by the one-thousandth of a grain. For almost every purpose of practical calculation, a decimal of three places is sufficiently exact; and to whatever extent it may be necessary or desirable to carry the subdivision of a unit—a foot, a mile, an ounce weight, &c.—it will be easy to find a decimal which, by increasing the number of figures, shall approach to the absolute value of the given fraction as nearly as we please. Suppose that in turning into a decimal, we divide the numerator with ciphers annexed, by the denominator, until the third figure of the quotient is reached, thus

17) 9000(⚫529

The decimal 529 is less than by 1700 of unity, and 17000 is not so great as 1000 or .001. The rules for operating on vulgar fractions show this to be the case. All the figures, then, beyond 529, which continued division would produce, though infinite in number, would not amount in value to so much as 001. If the fourth decimal figure be taken, will be still more nearly represented. •5294 differs from by 170000 or 55000, which is less than ⚫0001. Adding the fifth decimal figure we have 52941, which differs from by a fraction less than ⚫00001, and so on, each additional place of decimals giving a result which approaches more and more closely to the value of, although it is impossible to obtain a decimal that shall exactly equal that value.

Let it be required to convert 13 into a decimal that shall not differ from it by so much as 100000000 Annex eight ciphers to both terms of 18; then divide both numbers by the original denominator;

1900000000

100000000

1300000008 divided by 19, gives 68421052 one hundred-millionth of a unit.

Answer. 68421052, true to the

It must now be fully manifest, that whatever degree of exactness a calculation may demand it is unnecessary in practice to resort to vulgar fractions with the object of ensuring a more precise result than could be attained to by the use of interminate decimals, since a decimal is always to be found which shall express a given quantity within the smallest assignable limit of error.

In all instances of the reduction of vulgar fractions to decimals, it is advisable, if the work do not terminate of itself within a few figures, to carry the division a little beyond the extent which may be considered sufficiently exact for our purpose; and to increase by 1 the last figure we intend to retain, if the succeeding figure should amount to, or exceed 5.

Suppose, that in decimalizing, we have brought out the quotient as far as, ⚫7543859.

3.5

Now, suppose that we have no occasion for more than three places of this result. In that case, it would be improper to augment the third figure of 754-that is, to write, 755 instead of 754-as a compensation for omission of the remaining figures of the decimal, since 754 is nearer than 755 to the value of Both expressions are slightly in error. The first is too small, and the second too large. But, 22 the deficiency of the first, is not so great an error as the 67000' excess of the second. Therefore, if we mean to keep only three decimal places, 754 should be taken without any change. But if four places are to be retained, it would be more correct to increase the fourth figure by 1, than to adopt that part of the decimal as it stands. 7544, although in excess of the true value of 43, is not so much in excess as ⚫7543 is in defect. The first is too large by 70000

In dealing with a terminate decimal, such as 046875 (the exact equivalent of 3), which ends with a 5, it would be a matter of indifference if the last figure only were rejected, whether we wrote the preceding decimal as 04687, or as '04688. The error of defect on the one hand, would be the same as the error of excess on the other since the taking away of 000005 from 046875, or the adding of ⚫000005 to it, must equally remove it from the value of With this exception then, which need hardly be excluded from the general rule, it will ensure the greater accuracy of every decimal result, to increase by 1, the last figure that may be retained, when it is seen that the next figure would be 5 or greater than 5.

Practice of the Excise in decimal calculations.-In Excise calculations, it is seldom necessary to reduce a vulgar fraction to a decimal, or indeed, to operate in any way on vulgar fractions, except those of money, as all the measurements and estimates of quantity are directly expressed in decimal denominations of length, solid content, &c. A malt cistern, for instance, is measured in inches and tenths of an inch, and the area of it computed to the hundredth of a bushel; not, however, to the nearest hundredth which the dimensions would yield, if a corrrection were made for the thousandths' figure of the result, but only to the hundredth which is actually obtained on carrying the computation that far. The remaining decimal parts of a bushel are given in favour of the trader. Similarly, in reducing a number of bulk gallons of spirits to an equivalent at proof strength, the tenth of a gallon is taken as it is found: no regard is paid to the value of the subsequent decimals. This system, of course, operates against the trader when an allowance is deducted; still the loss either to him or to the Revenue is very trifling, even on the largest amounts. Besides, as the dimensions of vessels cannot be ascertained with ordinary gauging instruments to the precise tenth of an inch, and as the data for charging the duty on an article of variable bulk,-such as the assumption that every 811 bushels of barley put in steep, will increase to 100 bushels within a specified time—are themselves merely approximations to the general truth, it would be unfair to exact from the trader payment of duty on so minute a fraction as half a hundredth of a bushel or half a tenth of a gallon.

Repeating and Circulating Decimals.-This is a subject of no utility to the practical arithmetician, although interesting as a part of the general theory of decimal fractions; nor have officers been as yet required to exhibit any knowledge of it in their examination papers. It will be sufficient here to state, that every interminate decimal which results from the division of the numerator of a vulgar fraction by its denominator necessarily consists of periods of one or more figures which recur in the same order. Thus, 1111, &c. In this the repeating or recurring period has only one figure. 148148, &c., where the period has The question generally proposed

three figures; and similarly in other instances. in connection with this subject is, Given the period of a recurring decimal to find the vulgar fraction from which it was derived. All such decimals may be operated on by taking as many places as will serve the purpose of the computer, without any regard to the repetition of the figures.

Addition and Subtraction of Decimals.-RULE.-Reduce the decimals to a common denominator, if necessary, by annexing ciphers, so that all may have the

same number of places; then add or subtract as in whole numbers, and place the decimal point of the sum or difference, below the other points.

To save trouble, it is customary to omit the process of equalising the number of decimal places, and merely to conceive ciphers supplied; but care should be taken to arrange the decimal points and figures of the same local value in a vertical line under each other.

Examples: (1.) Add together 3527, 62 013, 002, and ⚫5.

These when reduced to a common denominator, are ·3527, 62-0133, 0020, and •5000, and proceeding as the rule directs,

The method of adding or subtracting decimals is the same in principle as that of adding or subtracting vulgar fractions: the fractions in either case are first brought to a common denominator, and their sum or difference is then found and placed over the common denominator. Thus, 0042+601·0042+ ·6010=10000t 60106052 = .6052,

10000

10000

Multiplication of Decimals.-RULE.-Multiply the given decimals as if they were whole numbers, disregarding all ciphers before the first significant figure, and from the right of the product point off as many places of decimals as there are in the multiplier and multiplicand together, making up this number, if necessary, with ciphers. Examples: (1.)

(2.)

(3.) 14.072

In Example (1,) as there are 8 places in the two factors, and as the product consists of 8 figures, all these are pointed off as decimals.

In Example (2,) as there are 9 places in both factors, and only 6 figures in the product, three ciphers are prefixed to it, to make up the requisite number of decimal places.

In Example (3,) three places are pointed off from the right of the product, because in both factors there is but that number of places.

The principle of the rule for valuing the product of two decimals is almost self-evident. Take as an illustration, example (2.)

lo multiply any Decimal by 10, 100, 1000, &c., it is only necessary to remove the decimal point as many places to the right hand, as there are ciphers in the multiplier.

Examples:

(1.) 016 multiplied by 10.16; by 100=1.6; by 1000=16, &c.

(2.) 32.579 multiplied by 10-325-79; by 100-3257.9 by 1000=32579, &c. The principle of this is too obvious to require illustration.

Contracted Multiplication of Decimals.-In most cases of the multiplication together of decimals for practical purposes, it is useless to go through the work in full, that is, to obtain all the decimal figures of the product, as the exactness of the result must depend on the exactness of the numbers which are multiplied by one another, and these are very rarely exact representations of quantity. There is generally an error of some amount in the last place, or the last few places of decimals, and the effect of multiplying figures which are in error, however slight, by other figures which are also in error, can only be to form a product which beyond a certain point is in greater error than either of its factors. If, for instance, in multiplying a length of 54-26 inches by a length of 38.79 inches, we are not sure that these dimensions are absolutely correct to the one hundredth of an inch, but that for anything we know, the second decimal figure should really be a unit more or less, it can be of no use to find a product of 4 decimal places, when it is uncertain that even the whole number resulting from the multiplication will be free from error.

As a general rule, it is useless to retain in the product any decimal places beyond those which are positively correct in its factors. For example, let the product of 72.563 and 9.154 be computed in the ordinary way

72.563

9.154

29 0252 362 815

725 63

65306 7 664-24 1702

Now, suppose that of this result, two decimal places only are required. In that case a great portion of the full-length process above shown is evidently superfluous, All that lies on the right of the vertical line should, if possible, be dispensed with, or, at least, only so much of it retained as will ensure the accuracy of the second