60 3+4+3=3+;;+83=17=211. 2. Take of a shilling from of a pound sterling. of a shilling of of a pound ? 20 Also of a pound 3 = 80 40 Hence 480 48.0 3. Find the difference between 125 and 83. = 18 480 Multiplication and Division of Fractions. RULE 1. To multiply a fraction by a whole number, multiply the numerator by that number, and retain the denomi nator. 2. To divide a fraction by a whole number, multiply the denominator by that number, and retain the numerator. 3. To multiply two or more fractions is the same as to take a fraction of a fraction; and is, therefore, effected by taking the product of the numerators for a new numerator, and of the denominators for a new denominator. (The product is evidently smaller than either factor when each is less than unity.) 4. To divide one fraction by another, invert the divisor, and proceed as in multiplication. (The quotient is always greater than the dividend when the divisor is less than unity.) 24 × 3 = 3 × == 2, and ÷ 3 = 3 × = 10° 3. Multiply £2 13s. 4d. by 31, and divide £4 15s. by 31. The embarrassment and loss of time occasioned by the computation of quantities expressed in vulgar or ordinary fractions, have inspired the idea of fixing the denominator so as to know what it is without actually expressing it. Hence originate two dispositions of number, decimal fractions and complex numbers. Of the latter, such, for example, as when we express lineal measures in yards, in feet (or thirds of a yard), and inches (or twelfths of a foot), 'we shall treat after a few pages. We shall now treat of the former. Decimal fractions, or substantively, decimals, are fractions expressed as whole numbers, but whose values decrease from the place of units progressively to the right hand in the same decuple or tenfold proportion as the common scale of whole numbers increase to the left. They are usually separated from the integers by a dot placed between the upper part of the figures. Thus, 2276 expressed according to the decimal notation is 22.7. The value of a decimal fraction is not altered by ciphers on the right hand for 500, or, is in value the same as, or •5, that is. : When decimals terminate after a certain number of figures, they are called finite, as 125 = 125 1000 = ,958 = 958 1000 When one or more figures in the decimal become repeated, it is called a repeating or circulating decimal; as .333333, &c. =,66666, &c., 428571428571, &c. = , and many others. Rules for the management of this latter kind of decimals are given by several authors; but, in general, it is more simple and commodious to perform the requisite operations by means of the equivalent vulgar fractions, from which circulating decimals are educed. Reduction of Decimals Reduction of Decimals is a rule by which the known parts of given integers are converted into equivalent decimals, and vice versa. Case 1. To reduce a given vulgar fraction to an equivalent decimal. Annex ciphers to the numerator, divide by the denominator, and the quotient will be the decimal required These two are evidently circulating decimals, in the former of which the figures 148 become indefinitely repeated, in the latter the figures 174603. 3. Reduce 14s. 6d. to the decimal of a pound. First 14s. 6d. = 1 Then 23 = 7.25 10 + 1 of 20 29 = 28+ = 28. 725, the decimal required. 4. Reduce to its equivalent decimal. 399 410 399 110 57 530 513 170 114 56 Note. The above fraction is = , of which the two denominators are both prime numbers (that is, divisible by no other number than unity), the entire equivalent decimal is a circulator of 18 places, i. e. one less than the last prime... 771929824561403508, 7719, &c. over again ad infinitum.* There are many curious properties of fractions whose denominators are prime numbers, one of which may be here shown in reference to fractions having the denominator 7. The circulating figures of the equivalent decimals are precisely the same, for 4, 4, &c. and in the same order: the circulate merely commences at a different place for each numerator. = 14285714, &c. = 28571428, &c. = .42857142, &c. Case 2.-Any decimal being given to find its equivalent. vulgar fraction; or to express its value by integers of lower denominations. When the equivalent vulgar fraction is required, place under the decimal as a denominator a unit with as many ciphers as there are figures in the proposed decimal; and let the fraction so constituted be reduced to its lowest terms. Or, if the value of the decimal be required in lower denominations, multiply the given decimal by the value of its integer in the next inferior order; and point off, from right to left, as many figures of the product as there were places in the given decimal. Multiply the decimal last pointed off by the value of its integer, in the next inferior order, pointing off the same number of decimals as before and thus continue the process to the lowest integer, or until the decimals cut off become all ciphers; then will the several numbers on the left of the separating points, together with the remaining decimal, if any, express the required value of the given decimal. Examples 1. Find the vulgar fractions equivalent to 25 and 375 •25= 25 10 1000 3 = , answers. 2. Find the value in shillings, &c. of 528125 of a £. |