; pressed as a complex fraction, as for instance g may be put into the form hence whatever may be the interpretation assigned to such a fraction as this, must be also assigned to every other fraction of the class, of which this is only a particular case. Now ž inch denotes a quantity composed of parts, of which 3 compose an inch, 2 of these parts being contained in the quantity, i.e. in other words, it denotes a quantity composed of parts, of which is compose an inch, of these parts being contained in the quantity. Hence we must interpret the meaning of such an expression as 5 inch to be, that it denotes a quantity composed of parts of which 54 compose an inch, 1š of these parts being contained in the quantity. Again į inch denotes a quantity such that, if the operation denoted by the denominator, viz. multiplication by 3 be performed upon it, it will become equal to the numerator; in other words, it denotes a quantity such that, if the operations denoted by * (which are equivalent to multiplication by 3) be performed upon it, it will become equal to the numerator. Hence we may interpret the meaning of the above-mentioned complex fraction to be, that it denotes a quantity such that, if 5% of it be taken, 13 inch will be formed. Now the operation by which such a quantity is found is that which has been defined to be division by the fractional denominator; 90 that such a fraction may be used to represent the quotient of any quantity by a fraction. Again: an abstract fraction as s is to be regarded as the symbol of division by 3 and of multiplication by 2 ; so also an abstract coniplex fraction must be regarded as the symbol of division of the unit by the denominator, and of multiplication by the numerator. Hence also it will denote the part, which the concrete units in the nnmerator are of those in the denominator; for evidently in order to form the quantity 13 inch from the quantity 54 inch, we must divide the latter by 5% and multiply it by 13, i.e. we 1 must perform the operations denoted by the symbol 57 Lastly: since 13 inch = 1 of 54 inch, therefore it follows that is and 57 1 13 tions which have to be performed upon the result of the operations denoted by 54, in order to produce the effect of performing those denoted by 19, i. e. it represents the quotient of the abstract fraction is by 54. Hence it appears that in all respects a complex fraction admits of the same interpretation as an ordinary fraction. We have now to deduce a Rule for the reduction of a complex to a simple fraction. Let the fraction already used serve as an example. According to the first explanation of the meaning of the fraction, we have to find a quantity of which 54 compose an unit. This is effected by the previous proposition by inverting the 53 5 of 5$ represent the same operation, therefore represents the opera divisor, and multiplying by it so inverted; doing this we find the quantity to be 3 of an unit; therefore the value of the fraction is 13 x 39 of an unit or 13 = 54 of an unit. This is the same result as we should obtain by using any of the other explanations given. Therefore in general to reduce a complex fraction, divide the numerator by the denominator. If the numerator and denominator be composed by the sum or difference of several fractions, it manifestly will not affect the result to bring them all to a C.D. And if this be done, the denominator will vanish in the divi. sion, and may therefore be omitted altogether. Henee the Rule 2 page 24 in the “ Practice." 9 Prop. 37.-To explain the system of notation of decimal fractions. In the ordinary system of notation, the number represented by a figure is always one-tenth of that which it would represent if it occupied the place of the figure on its left. Thus in the number 1111?, the 1 on the left denotes 10000; that next 1000, the next 100, the next 10, the last 1. Now if we place a point after this last I to shew that it is in the units' place, and then write a series of l's after it, the first of them should on the same principle represent one-tenth, the second one-hundredth, the third onethousandth, and so on, so that the first, second, third, &c. places to the right of the unit's place may be called the tenths', the hundredths', the thousandths', &c. places. Now every fraction with 10 or some power of 10 for its denominator may be resolved into the sum of a series of fractions having inferior powers of 10 for their denominators, every numerator being less than 10, i.e. may be resolved into the sum of a number of tenths, hundredths, &c. less than 10, and may therefore be denoted by writing the numerators in succession, supplying ciphers where any denomination of fraction is wanting, in order to keep the other figures in their proper places. 374 300 70 3 7 4 Thus + + + + =.374 7 6 = 24.706 2 4 + = .0024 10000 10000 10000 1000 10000 Hence it appears that in order to denote a fraction with any power of 10 as its denominator decimally, we must mark off by a point as many figures from the right of the numerator as there are ciphers in the denominator; and if there are not a sufficient number of figures we must supply ciphers on the left. + + = + 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 + + 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 so 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.-- T'o 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 one, 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 necessary. 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. Р The same Prop. exhibited algebraically:-Let and represent 10p 109 the vulgar fractions equivalent to two decimals containing respectively p and q decimal places; then q P Q РxQ PXQ Х 10P+9 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 mmber 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 visor, as whole numbers. P Q The same Prop. exhibited algebraically :-Let vulgar fractions equivalent to two decimals, containing respectively.p and q decimal places; suppose p to be greater than then P Q P 109 PQ Х 10p 109 10p 10p 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. 10p, 109, represent the 9 ; 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 period. Let the fraction be expressed in its lowest terms, so that there be no factor common to both numerator and denominator. The process, by |