Division of Repeating Decimals. 272. Find the fractional equivalents of the given numbers, divide as in vulgar fractions, being careful to cancel as much as possible, and reduce the quotient to decimals. If we have 9,2675 to divide by 2,59, we proceed thus: 9,2675÷2,59-92666 257: then 9999 92666 92666 101 × 257 25957 Examples. 1. 63,812,75 99 2. 289,0597 ÷,3647 = 798 3,569981+ 3. 53,296457 ÷,372=143,12677644+ Or, find the fractional equivalent of the divisor only; multiply the dividend by its denominator, and divide the product by the numerator. By this last method the scholar may prove any of the above examples. He will also see that it is, in many cases, the most convenient. 273. We obtain a convenient method of multiplying or dividing by 5, 5, 53, 54, &c., by finding the decimal value of a unit divided successively by each of those numbers, as in the following table: Now, if we divide any number by 5, it is evident that for every unit in it we shall have 2, which is the same as to multiply the number by ,2. Wherefore, to divide a number by any of the numbers in the left-hand column of the above table, we have only to multiply by the corresponding one in the righthand column. • Thus, to divide 76354 by 625, we have 76354,0016=122,1664, which is easily performed without even placing the numbers under each other. Again, the dividend 1, divided by the quotient ,2, must (60) give the divisor 5; consequently, any number divided by,2, will give 5 times that number. Wherefore, to multiply by any number in the left-hand column of the table, we have only to divide by the corresponding number on the right. Thus, to multiply 56479 by 3125, we have 56479,00032; which we perform by rendering the number of decimals equal in each, and suppressing the comma, thus: 3. 63792877 X 25=1594821925 9. 1135864279131253634765,69312 10. 11358642791 × 3125 = 35495758721875 The above examples may be proved by multiplying or dividing in the ordinary way. BOOK IV. TABLES OF MONEYS, WEIGHTS, AND MEASURES, WITH CALCULATIONS AND ILLUSTRATIONS OF TIME AND THE REGULATION OF CLOCKS-ASTRONOMICAL AND GEOGRAPHICAL MEASURE- REDUCTION, ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION OF COMPOUND NUMBERS-PRACTICE AND CALCULATION BY COMPLE MENTS -DUODECIMALS, BOARD MEASURE, &c. SECTION XIII. TABLES OF MONEYS, WEIGHTS, AND MEASURES, WITH CALCULATIONS AND ILLUSTRATIONS. 274. A number, when not applied to any particular species of quantity, is called abstract. Thus, when we say 3, or 3 times, the number 3 is abstract. When applied to a particular species, as when we say 3 books, 20 bushels, &c., it is called a concrete number. See Art. 18. Hitherto we have treated of numbers only as abstract; but we shall shortly consider their application to the measurement and valuation of quantities. We have already observed (3) that, to measure a quantity we must compare it with some known quantity of the same kind, which is called a unit. Now, as quantities differ in their nature and magnitude, the units or measures to which they are compared vary accordingly. 275. A number which is made up of units of different magnitudes is called a compound number.* Thus, 6 pounds 15 shillings and 6 pence is a compound number, because the pound differs from the shilling, and each of these differs from penny. the The following Tables will show what relation, or ratio, the *Perhaps, as some have suggested, it might more properly be called a complex number. 166 different units, by which we usually measure quantities, have to each other, 276. Accounts are kept in England and Ireland, as well as in Canada and several other British colonies,-and formerly were in the United States,-in pounds, shillings, pence, and farthings; but, though each of these denominations has always the same ratio to each of the others, the value of each in one country is by no means the same as in every other. English or Sterling Money. The pound sterling is marked £, the shilling s., the penny d., and sometimes the farthing qr.; these being the initial letters of the Latin words libra, solidus, denarius, and quadrans, which signify pound, shilling, penny, and farthing respectively. 4 farthings, gr., make. 12 pence, or 48 qrs,...... 20 shillings, 240d., or 960 grs... 21 shillings.. 5 shillings Farthings are usually written as fractions of a penny, thus: 277. In the United States accounts are kept in dollars and cents, and the coins issued by government are of gold, silver, and copper: a large proportion, however, of the actual specie currency is a mixture of Spanish, French, English, and other coins. The decimal division of the following Table was adopted by the Federal Government, on account of the great facility it affords in calculation: Federal Money. 10 mills, m., make.......... 1 cent, ct. 10 dimes, or 100 cents...... 1 dollar, $ 10 dollars.... 1 dime, d. 1 eagle, E. The copper coins are cent and half-cent. In 12345 mills, how many eagles? How many dollars? Dimes? Cents? Read the number in dollars, cents, and mills. What part of a cent is the figure 5? Read the number in dollars and cents. 278. The unit of each of the different denominations of Federal Money, from the highest to the lowest, being formed in the same manner as the unit of any order in an abstract number,—that is to say, of ten units of the next inferior denomination,—it is evident that the four cardinal* operations may be performed upon these, as upon abstract numbers. To find the sum of $29,37 and $156,257, we place units of the same order under each other, and add them as abstract numbers, thus: 29,37 156,257 185,627 * Cardinal, chief, principal. |