Conversely the sum of any number of fractions with different powers of 10 for their denominators, (the numerators being less than 10) may be expressed as one fraction with the highest power of 10 as its denominator, and for the numerator the number composed of the several numerators written together as one number. 3 7 4 300 70 4 374 Thus .374 =-+ + + + 24 + 10000 Hence it appears that in order to write a decimal as a vulgar fraction, we must take for the numerator the figures of the decimal as they stand, and for denominator 1 followed by as many ciphers as there are figures after the point. Cor. 1. Hence it appears that a cipher placed on the right of the figures of a decimal does not alter its value, since it may be regarded either as the addition of 0, or as the multiplication of numerator and denominator of the equivalent vulgar fraction by 10. But if we place a cipher immediately after the decimal point, it has the effect of division by 10; for by so doing we change all tenths into hundredths, all hundredths into thousandths, and 80 on; or we multiply by 10 the denominator of the equivalent vulgar fraction, i.e. we divide the whole by 10. Cor. 2. Hence decimals may be expressed with the same denominator by equalising the number of decimal places in them, by adding ciphers on the right. Prop. 38.-To explain the Rules for Addition and Sub traction of decimals. Decimals may be expressed with the same denominator by equalising the number of decimal places, ciphers being added on the right of those which require them (Cor. 2 Prop. 37). Being so expressed, the numerator of their sum or difference is obtained as in vulgar fractions by adding or subtracting the numerators of the fractions; the denominator of course remains the same; and the result is expressed as a decimal by marking off the required number of figures for decimals. In practice the ciphers are omitted in writing, but retained in significance by placing all the deci. mal points under one another, the ciphers being supplied mentally in forming the sum or difference of the numerators; as in the following example:Add together 5.2467, 98.305, 1.05. Full method 5.2467 Working method 5.2467 98.3050 98.305 1.0500 1.05 Sum 104.6017 Sum 104.6017 Prop. 39.-To prove the Rule for Multiplication of a Decimal by any power of 10. Suppose the decimal to be expressed as a vulgar fraction, then every multiplication by 10 will take away a cipher from the denominator, i.e. will have the effect of removing the decimal point one place to the right. If all the ciphers in the denominator should be exhausted, before the multiplication is complete, then evidently the process must be completed by adding ciphers to the numerator (Prop. 10). Hence the Rule: remove the decimal as many places to the right as there are ciphers in the multiplier, supplying ciphers if necessary on the right. Prop. 40.--To prove the Rule for Division of a Decimal by any power of 10. Suppose the decimal expressed as a vulgar fraction, then every division by 10 will increase the ciphers in the denominator by onę, i.e. will have the effect of removing the decimal point one place to the left. If there should not be any figures to the left of the point, then (Cor. 1 Prop. 37) ciphers must be supplied. Hence the Rule: remove the decimal point as many places to the left as there are ciphers in the denominator, supplying ciphers if pecessary Prop 41.-To prove the Rule for Multiplication of Decimals, Suppose the decimals expressed as vulgar fractions, then their product will be obtained by inultiplying all the numerators together for the numerator of the product, and all the denominators for the denominator of the product. But if different powers of 10 be multiplied together, there will be as many ciphers in the product as there are in all together: hence the denominator of the product will be ) followed by as many ciphers as there are in all the denominators, i.e. the product expressed as a decimal will have as many decimal places as there are in all the decimals multiplied. P Q The same Prop. exhibited algebraically:-Let and represent 10p 109 the vulgar fractions equivalent to two decimals containing respectively p and a decimal places; then P Q PXQ PXQ Х 10p 109 10P X 109 10P+4 Or the product of the two decimals may be found by multiplying the numbers together as whole numbers, and marking off p- q figures as decimals in the product. Prop. 42.-To prove the Rule for Division of Decimals. Since the addition of ciphers to the right of a decimal does not affect its value, let (if necessary) ciphers be added to the right of the dividend till the decimal places in it are greater in number than those in the divisor. Suppose now that the decimals are expressed as vulgar fractions, then the quotient may be obtained by dividing the numerator of the dividend by that of the divisor for the numerator, and the denominator of the dividend by that of the divisor for the denominator. But if one power of 10 be divided by another, the quotient will contain a number of ciphers equal to the difference of those in the two powers. Hence the denominator of the quotient will be 1 followed by as many ciphers as are equal in number to the difference between those in the dividend and divisor, i.e. the quotient expressed as a decimal will contain a mnmber of decimal places equal to the excess of those in the dividend over those in the divisor. And from above it appears that the figures of the quotient are obtained by dividing the dividend by the or, as whole numbers. P The same Prop. exhibited algebraically:-Let. 10p, 109 represent the vulgar fractions equivalent to two decimals, containing respectively p and 9 decimal places ; suppose p to be greater than 9, then -X 10 Or the quotient is obtained by dividing the dividend by the divisor as whole numbers, and marking off from the result a number of figures as decimals equal to the excess of those in the dividend over those in the divisor. Note. It is assumed in this Prop. that the number of decimal places in the dividend is greater than in the divisor, because, if this be not the case -originally, it may be made so. "Cor. Hence is apparent the method of converting 'a vulgar fraction into a decimal; for the numerator may be expressed as 'a decimal by adding ciphers after the decimal point, and then the division by the denominator may be perforined by the above rule. Prop. 43. To shew under what circumstances a vulgar fraction is convertible into a finite decimal; and that, in all cases, where the decimal is infinite, the figures recur in a certain order; and to find the extent of the re curring periodo Let the fraction be expressed in its lowest terms, so that there be no factor common to both numerator and denominator. The process, by 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 ý = 7X 10P Now in order that may be convertible 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 58: then a that is if be converted into a finite decimal, it will have r or s 2 x 58 decimal places, according as ro » or “ s. If b be of any other form than 2r X 5s its factors cannot be among those of a X 10p, therefore will not be convertible into a finite decimal. go In this case it is evident that, in dividing, every remainder must be less than b; therefore at the most there can only be b~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. 1 ..oi, 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. 1 3635 Thus .3635 = 3635 x .000i = 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 999 999 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 P or |