Ex. Divide 36 by 012. 36 = 36.000; and 36.000 divided by 012 is 3000, according to the rule. REDUCTION. 51. To reduce a vulgar fraction to a decimal. Add cyphers at pleasure, as decimals, in the numerator, and divide by the denominator according to the rule for the division of decimals. The truth of this rule is evident from Art. 11. 52. 33 In some cases, as in the last two examples, the vulgar fraction cannot exactly be made up of tenths, hundredths, &c. but the decimal will go on without ever coming to an end*, the * It is evident that no vulgar fraction can be exactly expressed by a decimal, unless it either has, or can be reduced to another which has, 10 or some power of 10, for its denominator, (Art. 39). Thus, reverting to the Exs. of the last Art. each of which vulgar fractions is expressed decimally with perfect exactness. But the following Exs. viz. and since they cannot be expressed by equivalent 3' 33' fractions with a denominator of 10 or some power of 10, are not capable of being expressed by terminating decimals. Also since 10, and its powers, are divisible by 2 and 5 only, it follows that no vulgar fraction can be expressed by a terminating decimal unless, when it is in its lowest terms, its denominator is divisible by one or both of the numbers, 2 and 5, and by no other number. same figure or figures recurring in the same order* ; but though we cannot represent the exact value of the vulgar fraction, yet, by increasing the number of decimal places, we may approach to it as near as we please. Thus 1 Now 1, or 1 1 11 is less than the true value by ; 11, or 10 90 100 41 Again =123123 &c., the figures 123 being repeated without 333 Decimals of this kind are called recurring or circulating decimals. Hence, although some vulgar fractions cannot be accurately represented by decimals, this affords no objection to the use of decimals, because for such fractions equivalent decimals can be found approximating to the true value as nearly as we please+. * This is easily shewn by a particular instance; and it may thence be seen to be 4 true in all cases. Thus, suppose it is required to find the decimal equivalent to 27 The required decimal is found by dividing 4.00000 &c. by 27; and if the quotient does not terminate, after each division there will be a remainder less than 27. Therefore, under the most unfavourable circumstances, at least after the quotient has reached to 26 figures, one of the remainders 1, 2, 3, &c....26 must recur; and consequently after that the figures in the quotient will recur. In the case proposed the remainder for one figure in the quotient is 13; for two figures, 22; for three figures, 4; which is a recurrence of the original figure: consequently the decimal is 0·148148, &c, the figures 148 being repeated in infinitum. Similarly also in other cases. † "The addition, subtraction, multiplication, and division, of decimal fractions, are much easier than those of common fractions; and though we cannot reduce all common fractions to decimals, yet we can find decimal fractions so near to each of them, that the error arising from using the decimal instead of the common fraction will not be perceptible. For example, if we suppose an inch to be divided into ten million of equal parts, one of those parts by itself will not be visible to the eye. Therefore, in finding a length, an error of a ten-millionth part of an inch is of no consequence, even where the finest measurement is necessary.' "In applying Arithmetic to practice, nothing can be measured so accurately as to be represented in numbers without any error whatever, whether it be length, weight, or any other species of magnitude. It is therefore unnecessary to use any other than decimal fractions; since, by means of them, any quantity may be represented with as much correctness as by any other method." De Morgan's Arithmetic, 3rd Ed. 131. 53. The method of reducing a terminating decimal to a vulgar fraction is pointed out in Art. 39. The following method will serve for converting recurring decimals into their equivalent vulgar fractions. and so on; where the recurring part of the decimal is always 1, preceded by as many ciphers as make the number of recurring digits equal to the number of 9's recurring in the denominator of the fraction. If then, for instance, the vulgar fraction equivalent to 0.1212 &c. be required, we have If the recurring period does not begin with the first figure after the decimal point, multiply and divide by such a power of 10 as will move the decimal point to the required position; then proceed as before. Thus, Similarly it may be shewn, that 0.09009009 &c. = 10 111 54. If it be sufficient for the purposes of any calculation to take a number of decimals less than the number given or obtained, the following rule is to be observed:— RULE. When the first of the figures struck off is 5 or >5, add 1 to the last remaining figure. Thus, if 2.7182818 be the decimal under consideration, 2-72 is nearer to the true value than 2.71, for 2.7182818-2.71, is 0.0082818; and 2.72-2-7182818 is 0.0017182, which is considerably less than the former difference. Also 2.7183 is nearer to the true value than 2.7182, as may be shewn in a similar manner. It may also be observed here, that in the multiplication of decimals some caution is requisite in taking the product as correct to a certain number of places of decimals, when either the multiplicand or multiplier is only approximately correct. Thus, if 3.12 express a certain length in inches, and is known to be correct within the thousandth part of an inch, the true length may be any thing between 3.12+ 1 1000 and 3.12 1 1000 that is, between 3.121 and 3.119; and if the proposed number is to be multiplied by 10, for example, the product is 31.2; whereas it may be any thing between 31-21 and 31.19, and therefore may not be correct even to one decimal place. 55. To find the value of a decimal of one denomination in terms of a lower denomination. This may be done by the rule laid down in Art. 27. Ex. Required the value of 615625 £. 56. To reduce a quantity to a decimal of a superior denomination. Divide the quantity by the number of integers of its denomination contained in one of the superior denomination, and the quotient is the decimal required. Ex. 1. What decimal of a shilling is threepence? 12) 3·00 25 Ans. For in the denomination shillings its numerical value must be of its value in the denomination pence. 1 12 First, we find what decimal of a penny 3 is; this, by the ule, is 75; then, what decimal of a shilling 4. 3 or 475 d. is; this is found in the same manner to be 3958333 &c.; lastly, we find, by the same rule, what decimal of a pound 13.3958333 &c. sh. is; which appears to be 66979166 &c. The conclusion will be the same if we reduce the quantity to a vulgar fraction (Art. 28), and this fraction to a decimal (Art. 51.) 57. It will often happen in practice that a whole series of vulgar fractions, instead of a single one, is to be reduced to a decimal, and in such cases considerable trouble may frequently be saved by making each fraction, when reduced, subservient to the reduction of some one or more of the others. Thus, Ex. 1. Required to reduce to a single decimal having 5 decimal places the following series of fractions: |