From these latter examples it will be observed, that the value of a decimal is not altered by annexing cy. phers; for 1 tenth is the same as 10 hundredths and 2 tenthsis the same as 200 thousandths, the significant figure, being just as far from the units' place in one case as in the other. But, as has been observed above, cyphers prefixed, (between the unit's place and the significant, figures,) materially affect the value, since every cypher, thus prefixed, removes each significant figure, one place farther from the units' place, and, of course diminishes its value ten fold. As these principles are important, we state them concisely as follows, I. CYPHERS ON THE RIGHT OF A DECIMAL DO NOT ALTER THE VALUE. But II. EACH CYPHER ON THE LEFT DECREASES THE VALUE TEN FOLD. Thus. .5, .50, .500, .5000, and .50000, are all equal in value, being each equal, as is evident to. But .5, .05, .005, .0005, .00005, and .000005 decrease in ten fold proportion. The pupil will now see the use of the point or period, which we have employed to separate decimals from whole numbers. If he were required to read this decimal, .15, he would call it 15 hundredths, but if there were no point, as here, 15, he would call it 15 (units or whole numbers,) simply. The point, then, enables him to determine what name and what value belongs to a decimal, by showing him its distance from the units' place. As it separates decimals from whole numbers, it is usually called a SEPARATRIX. The following examples contain whole numbers and decimals mixed, and are therefore called MIXED NUMBERS. Three, and five tenths Four, and seventy four hundredths 8.93 7.625 9.842 7.6324 5.55555. 3.5 82.594 90.006 825.0304 9004.010203. 827.34359. NOTE. When speaking of Federal Money, we remarked that, dollars and dimes might all be read as dimes; dollars, dimes, and cents might all be read as cents; and dollars, dimes, cents, and mills might all be read as mills. The reason of this is that Federal Money increases in ten fold proportion, in the same manner as decimals. For the same reason whole numbers and tenths may all be read together as tenths; whole numbers, tenths and hundredths may all be read together as hundredths; and so on to any distance. Thus 2.7 may be read two, and seven tenths, or twenty seven tenths; 3.25 may be read three and twenty five hundredths, or three hundred and twenty five hundredths, &c. The pupil will perhaps be ready to demand, what are decimals but Fractions. Fractions (§ xxvI.) are ex pressions for parts of numbers, and so are decimals. We reply that decimals ARE Fractions, and are often called DECIMAL FRACTIONS. The only differerence between them and any other Fractions is, that they increase and decrease just like whole numbers, by tens, and therefore we are not obliged to write down their denominators. If we please, however, we may write the denominators, and this we shall easily do, from observing the name of the decimal. Thus, 2.5=25.75=7%. 2793485, and so on. 100000 .8639. It will be seen by these examples that, 27.93485= THE DENOMINATOR OF A DECIMAL ALWAYS CONSISTS Of a unit, or 1, WITH CYPHERS ANNEXED. Fractions having other denominators, are called VULGAR FRACIt will be further seen by the above, that, TIONS. THE NUMBER OF CYPHERS IN THE DENOMINATOR OF A DECIMAL IS THE SAME AS THE NUMBER OF PLACES BELONGING TO THE DECIMAL. Or, the distance of the lowest figure of the decimal from the separatrix, determines the number of cyphers in the denominator. When the denominator is actually written, cyphers on the left of the decimal may be neglected; but when it is written, and afterwards removed, the cyphers must of course be carefully restored. In the following, let the denominators be removed, and the numerators writen "decimally, 21. 891056 100000000 13. 194,13 14. 271000000 When decimals are written, with their denominators, like Vulgar Fractions, it often happens that they can be Let the following be thus writ reduced to lower terms. ten and reduced as low as possible. NOTE. We have mentioned Federal Money as being similar to decimals. In fact, the denominations of Federal Money are decimals. The dollar is considered the unit or whole number; dimes stand in the tenths' place, cents in the hun. dredths', and mills in the thousandths'. The French weights, measures, &c., we have likewise seen are on the principles of decimals. (§ XXXIX.) The advantage in the use of decimals consists in the uniformity of their denominators, and in their regular increase by 10 like whole numbers. Hence, we have always to carry for 10; and the operations of the four ground rules are as simple as in whole numbers. $ LIX. 1. Change a dollar to a decimal. Here we wish to find how many tenths, hundredths, &c. of a dollar, there are in a dollar. Now a tenth of a dollar is a dime; hence, we have to inquire how many dimes there are in a dollar. a dollar is 10 times 2 a dime, because 10 dimes make a dollar. dollar is of a dime=5 dimes=$0.5 Ans. 2. Change to a decimal. Then a This question is the same as the last, except that it is not Federal Money. We wish here to find how many tenths, hundredths, &c. there are in a unit. a unit A. $0.125. is 10 times a tenth,=. of a tenth,=.5 Ans. A. .0625 NOTE. If we multiply by 10, to reduce the Fraction to tenths, it becomes of a tenth, which not being a whole tenth, we are obliged to place a cypher in the tenths' place, and proceed as before. 6. Change to a decimal. A. .2 NOTE. It is best to reduce the vulgar Fraction, first, to its lowest terms. 7. Change 8. Change 9. Change to a decimal. A. .2 to a decimal. A. .5 I, 45' 10. Change,, and, 27 A. 1600 .25, .625, .2 and .1875 to decimals. A. .025, .05 and .01 11. Change, 340, 356, and to decimals. 144 15 8 to decimals. 12. Change, o, and Another mode of explaining the above operation may be given. 13. Change to a decimal. If both numerator and denominator of a Fraction be multiplied by the same number, the value will not be altered, (§ XLI.) I may therefore multiply both the 7 and the 16 in the above Fraction, by 10, 100, 1,000 or any other number, without altering the value. Multiply both, then, by 10,000 and the Fraction becomes -70000. Now (§ XLI.) I may divide both numerator and denominator by the same number without altering the value. Divide them both then, by 16, and the Fraction becomes 435=(§ LVIII.) .4375 Ans. 0000 Either mode of explanation gives us the following rule. ANNEX CYPHERS TO THE NUMERATOR AND DIVIDE BY THE DENOMINATOR. THE DECIMAL WILL CONSIST OF AS MANY PLACES AS THERE ARE CYPHERS ANNEXED. NOTE. If there are not as many quotient figures as the rule requires, prefix cyphers enough to make out the number. An improper Fraction must, of course, be first reduced to a mixed number. 14. Reduce to a decimal. 25 A. .0016 15. Reduce to a decimal. A. .028 230 16. Reduce to a decimal. 17. Reduce 0 A. .05625 NOTE. We see here, that we may go on forever, and the decimal will continue to repeat 33, &c. 18. Reduce to a decimal. A. .18181818181818+ NOTE. This decimal goes on, repeating like the other; but it repeats two figures, 18, instead of one, as before." Decimals, which continually repeat the same figures, are called REPEATING DECIMALS, or REPETENDS. When one figure is repeated, the decimal is called a SINGLE REPETEND; when two or more, à COMPOUND REPETEND. Repeating decimals are also called CIRCULATES, or CIRCULATING DECIMALS. Properly, the term circulate, belongs to compound repetends. When other decimal figures precede the repetend, it is called a MIXED REPETEND; when otherwise, a PURE, or SIMPLE REPETEND. Repetends are also called INFINITE DECIMALS; decimals which terminate, or come to an end, are called FINITE. In single repetends, the repeating figure is commonly written only once, with a point over it, thus, .3. In compound repetends, all the repeating figures are once written, and a point placed over both the first and last, thus, .18. This notation shows us all the figures that repeat, and if it is necessary to extend the decimal lower, for the purposes of Addition, Multiplication, &c. we can write it down immediately, without the trouble of calculation. For further particulars on this subject sce § LXIX. 19. Change to a decimal. A. .142857 20. Change to a decimal. A. .6 21. Reduce 123 to a decimal. 22. Reduce 1 to a decimal. 11 A. .008 A. .6875 23. Reduce 17 to a decimal. A. .85 20 24. Reduce to a decimal. A. .03125 32 25, Reduce to a decimal. A. .037 26. Reduce to a decimal. A. .0384615 3, 27. Reduce, fs, 44, 7, 36, 375, odo,, and to decimals. LX. 1. Reduce 1 pt. to the decimal of a gallon. Ans. 1 pt. gal.=.125 2. Reduce 9 hours to the decimal of a day. Ans. 9h. day=.375 3. Reduce 2 ft. 6 in. to the decimal of a yard. Ans. 2ft. 6in. 30in.=38=yd.=.83 4. Reduce 5 fur. 16 rds. to the decimal of a mile. Ans. .675 5. Reduce 12s. 6d. 3 qrs. to the decimal of a pound. Ans. .628125 NOTE. When decimals, whether finite or infinite run on to a great number of places, it will usually be sufficient to take them to three or four, and neglect the rest. For the lower decimals become so small that they are of little importance. If on carrying the decimal as many places as you wish, the remainder left is less than half the divisor, the decimal, as it stands will be sufficiently accurate; if it be more, increase the last figure of the decimal by 1. It will be seen that neither of these decimals will be perfectly accurate, the former, being too small, and the latter too large. When a decimal is thus taken too small, the signis usually annexed to it, signifying that the true decimal is more. When it is taken too large,. the sign is annexed, signifying that the true decimal is less. Decimals written with these signs are called APPROXIMATES, because they are only an approximation to the truth. 6. Reduce 47£. 16s. 73 d. to a decimal expression. A. £47.8322916 NOTE. It would be sufficient to say £47.832+. We have carried the deci mals several places, throughout these examples, for the purpose of testing the accuracy of the pupil. 7. Reduce 83£; 19; 51⁄2 to a decimal. Ans. £83.972916 or £83.973-—. 8. Reduce 2 gals. to the decimal of a hhd. Ans. 031746 or 032 9. Reduce 15s. 9d. 3 qrs. to the decimal of a pound. Ans. .790625 or .79+ or .791—. For practical purposes, there is a shorter mode of reducing shillings, pence and farthings to the decimal of a pound. It is not perfectly accurate, but sufficiently so, in most cases. Take the last example. As 1 shilling is of a pound, so 2 shillings are of a pound. For every 2 shillings, then, we have 1 tenth |