MATHEMATICS. ARITHMETIC. 



[LONG DIVISION. 



the following statement. ex| 

 fora* of notation here adverted to: 



2185095 + 3-728365. 

 ) 8306184 + 8-276023. 



47061412 + 0- 7l l -''i 



- 115637. 



>d in ana or other of th 



^- 6528302 ,V 



It is plain that division by 10 requires no work : the 

 quotient is always the dividend itself, wanting the lasl 

 figure, which is the remainder, and which, therefore, 

 written as in this hut example, with the divisor under- 

 neath, completes the quotient. In a similar way, to 

 divide by 100 we have simply to cut off two figures from 

 the dividend for remainder ; to divide by 1000, to cut 

 off three figures ; and so on : thus 

 7<.ir, 



KM. - "1000 



All this is obvious, because 



78546-78500 + 46=78000 + 546, &o. 



II. When the Divisor is greater than 12. 



RCLK. Place the divisor to the left of the dividend 

 as in the former case, and to the right mark off a place 

 for the figures of the quotient. 



Find how many times the leading figure of the divisor 

 is contained in that of the dividend, or in the number 

 expressed by the first two figures, if the leading figure 

 of the dividend be smaller than that of the divisor ; and 

 I 'lit the figure expressing the number of times in the 

 quotient's place. 



Multiply the divisor by this first quotient-figure, and 

 subtract the product from the number formed by the 

 leading figures of the dividend, and to the remainder 

 annex the next figure of the dividend. The number 

 thus formed will be a new dividend, and the number of 

 times it contains the divisor to be found as before 

 will be the second quotient-figure, the product of which 

 and the divisor, being subtracted from the new dividend, 

 will give a second remainder, to which the next figure of 

 the original dividend is to be joined, and the operation 

 continued till all the figures of the dividend have been 

 u-. d. 



An example worked at length will explain the opera- 

 tion better than any verbal rule. 



Let it be required to divide 256438 by 346. 

 Placing the divisor on the left of 

 the dividend, and marking off a place 

 for the quotient on the right, we look 

 at the l/iing figure of the divisor 

 and also at that of the dividend, 

 with the view of seeing whether the 

 Utter contains the former, which it 

 does not, 3 being greater than 2 : we 

 therefore commence with the number 

 25, formed by the first two figures of 

 the dividend, and seeing that 3 is 

 mUined in 25, 8 times, we should 

 put 8 for the first quotient-figure ; but bearing in mind, 

 that when the whole divisor u multiplied by this 8, we 

 must attend to the carryings, we perceive that 8 is too 

 great, we therefore try 7, and find 7 times 346 to be 

 -'-% a number leu than 2664 above it, to that we can 

 I.IH-V the direction of the rule and subtract: the re- 

 mainder U 142, which, when the next figure of the 

 dividend is brought down, becomes 1423. We now take 

 thit as a dividend ; and, looking only at leading figures 

 in this new dividend and in the divisor, we see that the 



346)256438(741 

 2422 



1423 

 1384 



398 

 346 



62 



latter iriW go, as it is called, 4 times ; w therefore put 

 4 for the second quotient-figure ; and multiplying and 

 subtracting, we get 39 for the second remainder ; and, 

 by bringing down another figure, 398 for a new divi- 

 dend : the divisor goes into this once; so that the 

 quotient U 741, and the final remainder 52 : this re- 

 mainder, as in the former case, must be annexed, with 

 the divisor underneath, to the quotient-figures ; so that 

 the complete quotient is 741-^, which is the 346th part 

 of 256438. Of the truth of this you may convince your- 

 self by observing that 256438 has been cut up into 

 portions, and the 346th part of each portion found ; for 

 the work above is nothing else but 

 that here annexed, with useless re- 

 petitions suppressed. According to 

 this arrangement it is at once seen 

 that 700 is the 346th part of 242200, 

 that 40 is the 346th part of 13840, 

 and that 1 is the 346th part of 346, 

 and that of 52, the 346th part, still 

 remains to be taken. Now, 242200 + 

 13840 + 346 + 52 = 256438; conse- 

 quently 741, together with 5*,^, is the 



346th part of the number proposed. 

 It must be noticed that if any 



dividend, formed by a remainder and 



a figure brought down, should be If is 



than the divisor, that the divisor will 



go no times in that dividend ; so that 



a will be the corresponding quo- 

 tient-figure ; and that then a second 



figure must be brought down, as in 



the operation here annexed ; where 



the complete quotient is IOC -*fe. 

 There is another thing also 'to be 



attended to. Sometimes the divisor 



ends with zeros or noughts: when 



such is the case, the best way is to 



cut the ciphers off, and entirely to 



disregard them in the division, cutting 



off, however, at the same time, as 



many figures from the end of the 



dividend, which latter figures help 



io form the final remainder : you will 



see by operating on the same ex- 

 ample first with the ciphers retained, 



and then with the ciphers dismissed, 

 hat nothing is omitted but useless 



ciphers : the complete quotient being, by either way, 

 In thus completing the quotient, by means of 



.ho. final remainder, take care to restore the ciphers 



.hat were temporarily cut off from the divisor : in some 

 >oks on arithmetic this has not been named. Thu 



'ollowing examples are subjoined for practice : 

 13301 



346)256438(700 

 242200 



346) 14238(40 

 13840 



346) 398(1 

 346 



62 



472)48165(102 

 472 



965 

 944 



21 



2700)164826(61 

 16200 



2826 

 2700 



Rem. 126 

 27.00)1648.26(01 



lisa 



28 

 ' 27 



Rem. 126 



47 



159894 + 658 

 278C43 



35 



7S<i!r,7+3700 

 36326599 



(1.) 



(2.) 

 (3.) 



(*) 

 (5.) 



(6. ) 3939040647 + 6889 = 671787^^ 

 'o prove whether the quotient, in any case, is correct, 

 t is only necessary to multiply it and 

 ho divisor together : the product will 472 



)e the dividend, if the operation lie cor- 102 



ect, as is obvious ; for the object of 



ivision is to find a number which by tilt 



x multiplication with the divisor shall 472 



equal the dividend. In thus proving 21 Rem. 



ivision, if there be a renuiim/cr, this 



to bo added to the product of the 48166 



ivisor and quotient-figures. For ex- 

 _ilo, to prove whether the work above is correct, 



we multiply and add as in the margin ; and as the re nit 



is the name as the dividend, we may be sure that the 



quotient is t 



