Page:Encyclopædia Britannica, first edition - Volume I, A-B.pdf/484

 402 A R I T H M E T I C K. Ex. Reduce to an improper vulgar fra&ion. Rule. Set afide by a comma on the left, as many figures as there are places in the longed finite part, and 8vT8 then prolong the feveral circles to as many places as will make them fimilar. Num. 75 Ex. To make .54,63, and .54>63, = .54,636363, = V —8 T .9,148, conterminous. .9,148, = .91,481481, Den. 9 Here, becaufe .54, the Approximate decimals being imperfeft, cannot be ex- longed finite part, confids of two places, fet afide .91, actly reduced back to the vulgar fradtions from which they in the other circulate, for a finite part, and then prorefulted. But if the approximate be completed by an- long both circles to fix places, which renders them fimilar. nexing to it a vulgar fradtion, whereof the remainder of the divifion is the numerator, and the divifor the deIII. Addition of Decimals. nominator, you ftiall have a mixt number, which you I. Place the given decimals fo that the points may reduce to an improper vulgar fradtion ; then to mayRule dand diredtly under each other, and confequently the denominator annex as many ciphers as there are tenths, hundredths under hundredths <bc, ; figures in the approximate ; and this fradtion reduced tenths ifunder the given decimals be all finite or approximate, to its lowed terms, will be the primitive vulgar fradtion then, add. them as integers, inferting the decimal point direftly required. under the column of points. The figures on the left of Phob. VI. To reduce unlike circles to others that the point are integers, and thofe on the right are a deciare fimilar and conterminous. of the integer, confiding of as many places as there Similar or like circles are fuch as confift of an equal mal are figures in the longed of the given decimals. number of places. The operation is the fame here as in Ex. Thus, .27, and .09, are fimilar circles, as confid- addition vulgar fractions; for a cipher ing of two places each. But .63, and .148, are unlike; the ofright of a decimal does not the former confiding of two, and the latter of three onchange its value: If, therefore, ciphers places. fo as to give every.decimal Conterminous circles are fuch as begin and end at the bethe annexed, Time number of places, as is done in fame didance from the decimal point. the margin, they will by this means be Thus, .153846, and .384615, are conterminous ; be- reduced 'caufe they both begin at the place of primes, and have viz. 1000.to a common denominator, 3.495 3TWi3 an equal number of places. And .0,714285, and Note, If the decimals to be added are of different de.7,85714:2, are conterminous, becaufe they both begin at fird reduce them to one denomination, and the place of feconds, and have the fame number of places. nominations, add. The reafon is, becaufe like things only can But .81, and .1,36, are not conterminous, the former bethenadded or fubtraded. beginning at the place of primes, and the latter at the Ex. What the fum of .7251. and .625 s. ? place of feconds. Again, .63, and .481, are not con- Here you mayis either reduce the decimal of a fhilling terminous, becaufe they have not the fame number of to that of a pound, or you may reduce the decimal of a places ; for circles cannot be conterminous unlefs they pound to that of a fhilling. be at the fame time fimilar. reduce the decimal of a fhilling to that of a pound, Unlike circles are reduced to fimilar ones by the fol- by Fird /eduftion-afcending, viz. divide by 20, as follows. lowing Rule. Find the lead multiple of the numbers de2o).6250o(.03125 noting the number of places in the feveral given circles, .725 ^ and extend each of the given circles to as many places as there are units in the lead multiple. Sum .75625 = 15 li Thus, to reduce the unlike circles .63, = .636363, .63, and .148, to fimilar ones, ex- Secondly, reduce the decimal of a pound to that of a .148, = .148148, tend both circles to fix places, be- fhilling, by redu&ion-defcending; that is, multiply by caufe 6 is the lead multiple of 2 and 20, as follows. .725 1. 3, the number of places in the given circles. 20 In a circle any one of the circulating figures may be The anfwer here made the fird of the circle. Thus, 7.592, may be ex- is the fame as bepreffed thus, 7.5,925,; or thus, 7.59,259,; and that fore, without changing its value : confequently a pure circulate may put on the form of a mixt circulate, if one or more: figures on the left he fet afide for the finite part; thus,. .72,=.7,27, whereis.7,notis the finite may part. be thus demonThat the value changed drated. .7*27, = ^f-§ = -5-5-= .72. Hence two or more given circles may be made conterf. 2.0 minous, by the following AprRoxi"
 * . 15.1-; fum.