Page:Encyclopædia Britannica, first edition - Volume I, A-B.pdf/460

 378 A R I T H M E T I C K. you can have it 2 times; faying, 2 times 3 is 6 from IV If, as in the margin, 10, and 4 remains ; which 4 placed on the left hand of a cipher or ciphers, poflefs 648|o)89678|2(i the next figure 2 makes 42 : And again fay. Can I have the right hand of the divi6 alfo 2 times in 42 ? Atif. Yes; confequently 36 can for, cut them off, and cut 648 be had 2 times in 102; accordingly put 2 in the quo- off as many figures, viz. in tient, multiply and fubtraft; and to the remainder 30 this example, the figure 2 2487 bring down the next and laft figure of the dividend 6, from the right hand of the 1944 for a new dividual: Then, becaufe the dividual has a fi- dividend ; then divide the gure more than the divifor, fay. How often 3 in 30'? remaining figures of the diAnf. 9, and 3 remains; which 3 placed on the left hand vidend, viz. 89678, by the of the following figure 6 make 36: And again fay, Can remaining figures of the diI have 6 alfo 9 times in 36 ? Anf. No; confequently vifor, viz. 648, and you (2S42) 36 cannot be had 9 times in 306; therefore try if it can have the integral part of the be had 8 times, faying, 8 times 3 is 24 from 30, and 6 quot'Cnt; but to the remainder 254 annex the figure cut remains^ which IS placed on the left hand of the follow- off from the dividend, and you have 2542 for the nuing figure 6 makes 66: Again fay, Can I have 6 alfo 8 merator of your fraftion, and the whole divifor 6480 is times in 66? Anf Yes; confequently 36 can be had 8 the denominator. times in 306; wherefore put 8 in the quotient, andmul- The reafon will appear obvious by working a queftion in this manner, and alfo at full length, without cutting off the tiply and fubtradf as before: The lart remainder 18 is cipher or ciphers, and then comparing the two operations. the numerator of a fraction, and the divifor its denominator, to,be annexed to the integral part of the quotient; V. If, as in the margin, the 48|oo)978ojoo(203|.| figures cut off from the right as was taught in the former example. The preceding operation points out the manner of hand of the dividend, happen to 96 procedure when the divifor confifts of more figures than be all ciphers; in this cafe, the one, viz. you muft take the firft figure of the divifor out laft remainder, without regard180 of the firft figure of the dividual, or out of the firft two ing the ciphers cut off, is the 144 figures of the dividual in cafe the dividual have a figure numerator of your fradhon, and the fignificant figures of the dimore than the divifor: Then imagine the remainder to be "(36) prefixed to the next figure of the dividual, and try if you vifor the denominator. The can have the fecond figure of the divifor as often out of reafon is afligned in the do&rine of fradtions. this number; if you can, imagine again the remainder In like manner, if there be cut off from the dividend to be prefixed to the following figure of the dividual, any number of fignificant figures, with a cipher or ciand try if you can have the third figure of the divifor as phers on their right hand; in this cafe the laft remainder, often out of this number, foe.: but if you find you can- with the fignificant figures cut off, make the numerator not have fome fubfequent figure of the divifor fo often of your fradtion; and the fignificant figures of the dias you took the firft, you muft go back, and take the vifor, with as many ciphers as the number of fignificant cut off from the dividend, make the denomifirft figure of the divifor 1 time lefs, or fome number of figures times lefs out of the firft, or 6ut of the firft two figures nator. Thus, if, in the above example, the figures cut of the dividual: Then proceed as before, repeating the off from the dividend had been 50, the numerator of your trial till you find you have the fecond and all the fubfe- fradtion would have been 365, and the denominator 480. quent figures of the divifor as often as you took the firft. Coniranions in •working Divijion of Integers. But here obferve, that if, in trying how often the divifor can be had in the dividual, either 9, or a number 1. To divide any number by 10, too, 1000, foe. you greater than 9, any where remain, you may conclude, have only to point off for a remainder as many figures without further trial, that all the fubfequent figures of on the right hand of the dividend as the divifor has cithe divifor can be had as often as you took the firft; as phers, and the other figures on the left of the point or may be thus demonftrated. are the quotient; Thus, 7489634 divided Suppofe the fubfequent figures of the divifor to be the feparatrix by io, 100, 1000, foe. ftands as follows. highest poffible, that is, all 9’s, and the following fiQuot rem. gures of the dividual the loweft polfible, that is, all o’s ; 10)748963.4 again, imagine the remainder 9 prefixed to the follow100)74896.34 ing figure of the dividual o, that it will make 90 ; now 1000)7489.634 it is plain, that the fubfequent figure of the divifor 9 can 10000)748.9634 be had in 90, the higheft number of times pofTible, viz. 2. If the figures the divifor are ally’s, or all except q times, and 9 will remain; which prefixed to the next units figure, as 9,of99, 999, 98, 997^ 9996, foe. walk figure of the dividual o, makes 90, in which the fubfe- the as follows' quent figure of the divifor 9 can again be had 9 times, Find a new divifor, by annexing to unity as many ciand 9 will remain as before; therefore all the fubfequent as there are figures in the given divifor, fubtradl figures of the divifor can be had as often as you took the phers given frorrr the new divifor, and the remainder or firft ; ^and if they can be had in this cafe, much more can the difference is the complement. Divide the given dividend they be had when? a number greater than 9 remains. by the new divifor, viz. point off fa many figures on die