DECIMALS. DECIMAL fractions are those in which no other denominators are used than 10, 100, 1000, &c. and are expressed by writing the numerators only with a point or comma before it on the left hand, which is called the decimal point; the denominators are not written but intimated by the position of this point, for we understand as many ciphers following a unit in the denominator, as there are digits standing on the right hand of the decimal point. Thus, the decimal ,7 is equivalent to the vulgar fraction, the decimal ,37 with, and the decimal ,037 with T. A cipher placed to the right hand of a decimal, alters not its value, but if placed to the left hand of a decimal it decreases the value in a ten-fold proportion, for, as in notation of whole numbers, the values of figures increase in a ten-fold proportion from right to left; so on the contrary, in decimals, the values decrease from left to right, in the same ten-fold proportion, for,5,,50, ,500 are decimals of the same value, being each , that is, %, %%%; but every cipher annexed to the left hand of a decimal decreases its yalue to the tenth of what it was, before so placed, thus 5 is equal to +,,05 to and ,005 to you. Hence unity is considered as a fixed point, from which whole numbers proceed infinitely increasing, and decimals infinitely decreasing, as is farther explained by the following TABLE. Hunds. of Thousands. Thousands. Hundredths. Ten Thousandths. Thousandth Millioneth. Hund Millioneth. Tenths. Units. 9 8 7 6 5 4 3 2 1 ,2 3 4 5 6 7 8 9 Decimals, as well as whole numbers, decrease in a tenfold proportion towards the right hand; therefore, decimals have the same properties as whole numbers, and are subject to the same rules. Express in words the following decimals. ..7, ,841, ,073, ,8605, ,0005, 7419,,86402. 75,843, 85,684, 71008, 710,08, 7,1008, 7100,8 ,834, 07503, 17,0084. Express decimally, Three tenths. Five tenths one thousandth. Six thousandth. One tenth four thousandth.' Seven tenths, three thousandth and one millioneth. Nine tenths, eight hundredth, seven thousandth, REDUCTION OF DECIMALS, Reduction of Decimals is the converting of common fractional quantities and decimals into each other, the finding of the values of quantities decimally expressed, and expressing any given compound quantity decimally. CASE 1st. To reduce a common fraction to its equivalent decimal. Divide the numerator by the denominator, annexing ciphers as far as is necessary, the quotient will be the decimal required. CASE 2d. To express any decimal by a vulgar or common fraction. Under the given decimal place a unit, with as many ciphers annexed thereto, as there are places in the given decimal, and reduce the fraction thus formed to its lowest terms. Express,35 by a vulgar fraction. Express ,412 as a vulgar fraction. Answer. Answer. What vulgar fractions are equal to the following decimals, each respectively. CASE 3d. To find the value of a decimal in terms of the inferior denominations of the integer, of which it is a part. Multiply the decimal by the number of parts in the next inferior denomination; and cut off from the right-hand, as many places for decimals as were in the given decimal. Multiply the places cut off, by the parts in the next lower denomination, cutting off the first number of places as before, thus proceed until all the denominations are gone through, or until ciphers only remain, and the several parts separated on the left hand, will make up the answer. Examples. Required the value of £894. Required the val. of ,0625 cwt What are the values of the following decimals, each respectively? CASE 4th. To reduce numbers of divers denominations to their equivalent decimal values. Reduce the given quantity to a vulgar fraction, and this fraction reduce to a decimal 1. Reduce 13s 4d to the decl of a £. 2. Reduce 3qr 21tb to the decl of an ct Reduce each of the following compound quantities, to the decimals of the integer, of which they are respectively a part 11. 4qr lnl to the dec. of an ell E. 12. 13. 9d to the decimal of a £ 15. 104d to the decimal of a £ 14. 4oz 12dr to the dec of a b avr 2r 20p to the dec. of an acre. 16. 14 b to the decimal of a ton. CASE 5th. To find the value of the decimal of a pound sterling, by inspection only. *Double the first place or that of tenths for shillings adding 1 if the next place be 5 or more, then consider the figures in the second and third places, if the second was less than 5, or if greater, the remainder when 5 is subtracted therefrom, with the third place annexed thereto, as so many farthings, abating 1 when they exceed 12, and 2 when they exceed 37; the result will be the answer required. Or thus-Take one-fifth of the number expressed by the two first figures of the decimal for the shillings, the remainder, if any, with the third figure of the decimal annexed, diminished by a twenty-fifth part thereof, taken as farthings, will be the remaining part of the required value. Find the value of the following decimals of a pound, by CASE 6th.-To find the decimal of any part or compound parts of a sterl. by inspection.† Write half the greatest number of shillings for the first decimal place, and let the farthings in the given pence and farthings possess the second and third places, observing to increase the second place by 5, if the shillings be odd, and the third place by 1, if the farthings exceed 12, and by 2 if they exceed 37, or more accurately, by the 4th of the farthings in the given pence and farthings. Ten tenths or 20s make a pound, and 5 in the hundredth place is To or ', which is a shilling; also 960 farthings make a pound, which is less than 1000 by thereof, hence, both rules are manifest. This case is evidently the converse of the former, N |