ARITHMETIC. 



222 



121 



III 



105 



74 



operation is termed long division. For example, 

 let it be required to divide 23415 by 

 37. Here, arranging thedividend and 37)23415(632 

 divisor as in the margin, we begin 

 by trying to divide 2 of the divi- 

 dend, that is, 2 ten-thousandths, by 

 37. If we could perform this opera- 

 tion, the first figure in the quotient 

 would be tens of thousands, but we 

 cannot divide 2 by 37. We take in 

 the next figure of the dividend viz., 3 x 



3. This makes 23 ; that is, 23 

 thousands. If we can divide this by 37, then the 

 first figure in the quotient will denote thousands. 

 We cannot, however, divide 23 by 37, and hence 

 there are no thousands in the result We then 

 include the next figure in the dividend, that is, 4, 

 and try if we can divide 234, that is, 234 hundreds, 

 by 37. Now, on trial, we find that 37 is contained 

 in 234 6 times, with a remainder of 12; that is, 

 the first figure in the result is 6 hundreds, with 12 

 hundreds over. These 12 hundreds, or 120 tens, 

 taken along with the i ten of the dividend, give 

 i2i tens to be divided by 37, and this, we find, 

 will give 3, that is, 3 tens, with 10 tens over. These 

 10 tens, or 100 units, taken with the 5 units of the 

 dividend, give 105 units to be divided by 37. This 

 will give 2 units, with a remainder of 31 units. 

 The result will therefore be 632, with a remainder 

 of 31 units. These 31 units have yet to be divided 

 by 37, but this operation is merely indicated thus : 

 ff. So that the exact quotient will be 632^. 

 In the same manner all such divisions may be 

 performed. 



Division of Decimal Numbers. Let us now 

 divide a decimal number by another number. We 

 here distinguish two cases, and begin by con- 

 sidering the divisor first a whole number. For 

 example, let it be required to divide 3-1245 by 

 65 : this differs in no respect 

 from the division of whole 

 numbers. As before, we 

 arrange the dividend and 

 divisor as in the margin, and 

 begin by trying to divide the 

 3, that is, 3 units of the divi- 

 dend, by 65. We cannot do 

 this, and hence there are in 

 the quotient no units. We 

 include the next figure, which 

 is i, and divide, if we can, 

 31 tenths by 65. This we 

 cannot do, and hence there are no tenths in the 

 quotient. We include the next figure, that is, 2, 

 and try if we can divide 312 hundredths by 65. 

 This we can do, and we find that 312 contains 

 65 4 times ; hence the first figure in the result 

 denotes hundredths, and therefore it is written in 

 the second place from the point. The rest of the 

 division is performed as in the division of whole 

 numbers. 



Next, let the divisor be a decimal number, and 

 let it be required, for example, 

 to divide 6-3882 by 23-4. Now, 

 this is the same as dividing ten 

 times 6-3882 by ten times 23-4; 

 but ten times 6-3882 is 63-882, 

 and ten times 23-4 is 234 ; so 

 that, instead of dividing the given 

 dividend by the given divisor, we 

 will do what is exactly the same 



6s)3-i245(-o48o6... 

 260 



524 

 520 



450 

 39<> 

 ~~6o 



468 



1708 

 1638 



702 

 702 



thing, if we divide 63-882 by 234. By this device, 

 we have made the divisor a whole number, and 

 therefore, by the last case, we are enabled to per- 

 form the division as is here shewn. 



Again, let it be required to divide 6-3882 by 

 2-34. Now, in order to transform the divisor into 

 a whole number, it is sufficient 

 to multiply it by 100. We will 

 therefore divide 100 times the 

 dividend by 100 times the 

 divisor ; this will not alter the 

 required quotient. The work- 

 ing will evidently stand as in 

 the margin, and the first figure 

 in the quotient denotes units. 



234)638-82(2-73 

 468 



702 

 702 



468 



1708 

 1638 



702 

 702 



Again, let it be required to divide 6-3882 by 

 0234. This is the same as divid- 

 ing ten thousand times 6-3882 by 234)63882(273 

 ten thousand times -0234 that is, 

 it is the same as dividing 63882 

 by 234, which gives 273 as 

 quotient. 



These examples are sufficient 

 to shew how all such operations 

 are performed. 



The division of one number by another is often 

 indicated by putting the mark -i- between the 

 numbers to be divided ; thus, 8 -i- 4 is understood 

 to mean that 8 is divided by 4. This process is 

 more frequently indicated by writing one number 

 over the other, with a horizontal line between 

 them, the number to be divided being above ; 

 thus, f . 



Hitherto, we have considered only the arith- 

 metic of whole and decimal numbers ; we have 

 now to consider that of fractional numbers. But 

 before entering upon that part of our subject, it 

 is necessary to understand the following two im- 

 portant processes namely, that of finding the 

 greatest common divisor, and the least common 

 multiple of two or more numbers. 



Greatest Common Divisor Least Common Multiple. 



The greatest common divisor of two given num- 

 bers is the greatest number which divides each 

 without leaving a remainder. In finding the 

 greatest common divisor of two numbers, then, 

 we will consider two cases : first, when the one 

 number contains the other an exact number of 

 times. For example, let it be required to find the 

 greatest common divisor of 40 and 8. Now, it is 

 clear that no number greater than 8 will divide 8 ; 

 and therefore, as 8 divides 40, it is the number 

 required. Hence, in this case, the less of the 

 two numbers is that required. 



Again, let the one number not contain the other 

 an exact number of times ; for example, let it be 

 required to find the greatest common divisor of 

 14 and 35. It is evident that no number higher 

 than 14 will divide 14, but since 14 divides itself, 

 it must first be tried to see if it also 

 divides 35 ; if it does, it will be the 14)35(2 

 number required. On dividing 35 by 33 

 14, we find a remainder, 7. Although 

 this operation shews that 14 is not the 7 



number required, it yet puts us on the 

 way of finding it ; for whatever be the number 

 required, it is clear that it must divide twice 14 

 or 28. Then, as it also divides 35, it must divide 

 the difference between 35 and 28 that is, it must 



599 



