Page:Encyclopædia Britannica, first edition - Volume I, A-B.pdf/478

 $$$ ^ A R I T H l' r e t r c k. Or we may conceive the denominator of a decimal other fort of numerical computation; as will appear to be formed by the continual multiplication-of unity in the fequel. into as often as there—areioo,ciphers it. iaX Thus,10 i X io10,— to, and iXioXto and rXin10X II. Reduftion of Decimals. — xooo, &c. And becaufe the fraflions -j%,T%90-, fee. have the: highefl: numerators poffible, it is plain, Paob. I. To reduce a vulgar fraftion to a decimal. tliat the number of figures or placesdn the numerator of Rule. To the numerator of the vulgar fradtion affix point or comma, then annex a competent number of a decimal can never exceed the number of ciphers in the aciphers, and divide by the denominator; the quot as the denominator. It if ufual to write down only the numerator of a de- numerator of the decimal, and the cyphers annexedfhow l irtral fradhon^ omitting the denominator • and when the number of decimal places. the nhfner^tor has the lame nijmber of figures or places Examp. I. Reduce to a decimal ? as the denominator has ciphers, it is dope by writing Here to. the numerator 1 annex one cipher, 2)1.o(.5 dbwn the figures of the-numerator, and prefixing a point, and dividing by the denominator 2, the quot 10 to difiingui/h them from a whole number. So is writ- is 5, and o remains; and becaufe a fingle ci- — ten thus, .7; and is written thus, .2j. The point pher only was annexed to the numerator, the (o) decimal numerator will confift but of one fithus prefixed is caHed the decimal point. But when the numerator has-not lb many figures- or gure, namely 5 ; to which, therefore, prefix the deciplaces as there are ciphers ip the denominator, the de- mal point. So *=.5. fed is fupplied by prefixing a cipher for every, figure Hence appears the reafon of the rule ; namely, wasting, and then placing the decimal point on the left. 2 : 1 :: 10 : 5; that is, as the vulgar denominator to vulgar numerator, fo is the decimal denominator to So T^g is written thus, .03; and' thus, .0075 ; the and thus, .0005., the decimal numerator. From this manner of notation, it is eafy to read a Examp. II. Reduce ^ to a decimal. decimal, or to know its denominator,, viz. imagine 1 To the numerator 3, annex two ciphers ; 4)3 .o©(.75 to (land under the decimal point, and a cipher under and, dividing by the denominator, the quot 28 gives 75 for the numerator of the decimal,. every .decimal place. Thus, .9 is T®, and .48 is and; Sp.05- is and .007 is y-ggg, and .00036 is two ciphers having been annexed. So 20 i=-7520 Hencefromit the is plain, decimals, integers, creafe left to that the right, and like increafe from dethe' right to the left, in a decuple proportion. On the contrary, any decimal figure, by being removed one place, Though ciphers may be annexed at pleafure, yet it is the ciphers'ufed that: determine the number of detoward the left, becomes ten times greater. An integer, by annexing ciphers, is raifed to higher cimal places in the quot; and at firft it is fufficienti to annex fo many as ferve to complete the firft dividual, places op the left, and may by this means have its value leaving .room to annex more as you proceed in the opeir.creafed to infinity. On the other hand, a decimal, ration ; or rather annex the other ciphers to the remain* by prefixing ciphers, is depreffed to lower places on the ders, without giving them a place in the dividend. right, and may by this means have its value diminiftied The firft dividual alfo ffiows whether ciphers ought to infinity. Ciphers annexed to decimals do not change the va- to be prefixed to the quot, and how many.. Thus, i£ dividual take in only one of the annexed cilue of the decimals. Thus, .50=5, and .500=5, for the.firft the figure put in the quot is primes, and no ci.50= ^=^ and T4|?° ^=.5. parts, and phers, Decimals may=be.5;refolved into=conftituent pher to be prefixed. If the firft dividual comprehend of the annexed ciphers, the figure put in the quot the parts may be read, feparately, thus, .847 = 8 + two is feconds, and one cipher muft he prefixed. If the. .'04-f .007 =-r5the+figure t^o+T3^>’ dividual comprehend three of the. annexed ciphers,, In decimals next the point, being the firft firft figure put in the quot is thirds, and two ciphers muft decimal place, is fometimes called primes, and the fe- bethe prefixed, &c. Hence, in reducing a vulgar, fraccond figure from the point is called feconds, the next to a decimal, the natural and eafy way is, to place thirds, &c. Thus, in .875 the figure 8 is primes, 7 is tion firft the decimal point in the quot, and after it a cipher feconds, and 5 is thirds. From this brief account of the nature of decimals, or ciphers, or the quotient-figure, as the firft dividual it follows, that the manner of operation in decimals- direds. In reducing a vulgar fradion to a decimal, if o at laft will be the fame as in whole numbers ; and alfo, that remains, as in all the above examples, the'd'ecimal is the-fame number may be differently exprefied, accor- precifely equal vulgar fra&ion, and is called ay?ding as the integer is chofen. Thus, the time fince our nite or terminateto the Saviour’s birth may be written thus, 1769; or thus, Id finite decimals,decimal. the denominator is always fome a176.9; or thus, 17.69; or thus, 1.769; or thus, .1 769, liquot part, of the numerator increafed by annexing ciaccording as one year, a decad, a century, a chiliad, or myriad, is ufed as the integer. Hence arifes the phers; and fuch • decimals take their rife from vulgar fuperior excellency of decimal arithmetic, above every fradions whofe denominator is 2 or 5, or fome power2 orof