CHAMBERS'S INFORMATION FOR THE PEOPLE. 



Additional examples : 



742-3 7-423 



3.42 -342 



14846 

 29692 

 22269 



2538-666 



I 



I 



01 



14846 

 29692 

 22269 



2-538666 



ool 



LI 



I-I 



II 

 II 



I. .-I 



7423 



34-2 

 14846 

 29692 

 22269 



25.38666 



01 

 01 



-0001 



The multiplication of one number by another is 

 usually indicated by putting the mark X between 

 the numbers to be multiplied : thus 4 X 7 is un- 

 derstood to mean that 4 is multiplied by 7. 



DIVISION OF WHOLE AND DECIMAL NUMBERS. 



Division is that process by which we find 

 how often one number is contained in 

 another. For example, to divide 30 by 30 

 6 is to find how often 30 contains 6, 

 or how often 6 is contained in 30. We 

 can at once find how often 6 is con- 

 tained in 30 by subtracting 6 from 30 

 again and again, as often as we can, 

 and the number of subtractions we re- 

 quire to perform will mark the number of 

 times 30 contains 6. Here it is found 

 that 6 can be subtracted from 30 5 times ; 

 hence we say that 30 contains 6 5 times, 

 or 30 divided by 6 gives 5. The number 

 to be divided is called the dividend; the 

 number by which we divide is called the o 

 divisor; and the result of the operation is 

 called the quotient. 



All such operations may be performed in the 

 same manner ; but it is evident that when 

 the dividend is large compared with the 7524 

 divisor, the number of subtractions will 

 therefore be very numerous, and conse- 



quently the working will become very labo- 75 22 

 rious. For example, suppose we wish to find 

 by this method how often 7524 contains 2. jr 2o 

 If we begin by subtracting 2 again and 5 ~ 2 

 again, we see at once that the number of 

 subtractions is very great; and therefore 75 J 8 

 recourse must be had to some other method 

 to obtain the quotient, and that which con- 

 stitutes the rule of division is a short 

 method of obtaining the same result. Taking the 

 same example, we will begin by subtracting, not 2, 

 but 2000, and this we find we can do 3 

 times, with a remainder 1 524. But sub- 

 tracting 2000 3 times is the same as 2OOO 

 subtracting 2 3000 times. We have 

 yet 1524 to divide by 2 : instead of 55 2 4 

 subtracting 2, however, time after time, 



7524 



2OOO 



1524. 



- = 



from this number, we will subtract 100 3524 



times 2, or 200, as often as we can. 2000 



We shall evidently have 100 times fewer 



subtractions to perform : it will be 



found that 2do can be subtracted 7 



times with a remainder 124; but subtracting 



200 7 times is the same as subtracting 



598 



2 700 times. We have yet 124 

 to divide by 2 ; and instead of sub- 

 tracting 2 from this as often as we can, 

 we will subtract 10 times 2, or 20, as 

 often as we can, and this we find we can 

 do 6 times. But subtracting 20 6 times 

 is the same as subtracting 2 60 times. 

 There remains 4, which we must divide 

 by 2, and we find that from this number 

 we can subtract 2 2 times. Hence the 

 given number 7524 contains 2 3000 

 times, together with 700 times, together 

 with 60 times, together with 2 times ; 

 or, in all, it contains 2 3762 times. In 

 this way we may perform all such oper- 

 ations ; and the device which we here 

 introduce has enabled us to obtain the 

 result by means of 1 8 instead of 3762 

 different subtractions. In practice, how- 

 ever, it is usual to shorten even this 

 working, and the multiplication table 

 enables us to do so : thus, if we write 

 the dividend and divisor, as in the 

 margin, then 7, which is the 

 highest figure in the dividend, 

 and denotes thousands, is first 

 divided by 2. Now the multi- 

 plication table tells us that 2 times 3 

 makes 6, or that 7 contains 2 3 times ; 



1524- 

 200 



2 )75 2 4 



200 



1124 

 20O 



924 

 2OO 



724 

 2OO 



524 

 2OO 



324 

 200 



104 

 20 



84 



20 



64! 

 20 rl 



44 



20 



24 



20 



dividing 7000 by 2, therefore, gives 3000, 

 with looo over ; this 1000, or 10 hun- 

 dreds, taken with the next figure in the 

 dividend namely 5, gives 15 hundreds 

 to be divided by 2 ; this gives 7 hun- 

 dreds, with loo over ; and this 100, or 

 10 tens, taken with the 2 tens, gives 12 

 tens to be divided by 2 ; this gives 6 

 tens ; and the 4 units divided by 2 gives 

 2 units. Hence, as before, the quotient 

 is 3762. In the same way it is usual to 

 perform all divisions in which the divisor is less 

 than 9. 



Example 2. Let it be required to divide 37452 

 by 7. We write down the dividend and divisor 

 as in the margin, and begin by divid- 

 ing the 3 in the dividend, which 7)3741; 2 



denotes tens of thousands, by 7. We 



find we cannot do this, but we in- 535--- 2 

 elude the following figure 7, and 

 divide, if we can, 37 thousands by 7. This we 

 can do, and find, as a result, 5, with a remainder 

 2. This denotes 5 thousands, with a remainder 

 of 2 thousands, or 20 hundreds. These 20 hun- 

 dreds, taken with the 4 hundreds, gives 24 hun- 

 dreds to be divided by 7, and the multiplication 

 table tells us that 24 contains 7 3 times, with a 

 remainder 3 ; that is, 24 hundreds contain 7 300 

 times, with 3 hundreds over, or 30 tens. These 30 

 tens, taken along with the 5 tens of the dividend, 

 give 35 tens to be divided by 7, which gives 5 

 tens exactly. The 2 units now require to be 

 divided by 7. This cannot be done ; that is, 

 there are no units in the dividend, but a remainder 

 of 2. Hence the quotient is 5350, with a remain- 

 der of 2. 



When the divisor is less than 9, it is usual to dis- 

 pose the working as we have done in the examples 

 given above, and the operation is termed short 

 division : but when the divisor is above 9, or any 

 large number, it is usual to exhibit the working 

 as is shewn in the following example, and tl 



