CHAMBERS'S INFORMATION FOR THE PEOPLE. 



other numbers to be added : these are written 

 under each other, as in the margin, the 

 simple units under each other, the tens 453 

 under each other ; and so on. On adding the 

 simple units, we find their sum is 21 that 53'4 

 is, i simple unit and 2 tens ; the i unit is 



written under the units, and the 2 tens are 13521 

 taken and added along with the tens 

 when the tens are added, we find their sum is 12 

 that is, 2 tens and i hundred ; the 2 tens are written 

 under the tens, and the i hundred is taken and 

 added along with the hundreds of the given 

 number ; this makes their sum 1 5 that is, 5 hun- 

 dreds and i thousand, which, when added along 

 with the thousands of the given number, makes 13 : 

 hence the sum is 13521, or 13 thousands, 5 hun- 

 dreds, 2 tens, and i unit. 



This simple method is employed in adding 

 decimal numbers together. Thus, let it be required 

 to add 4-75, 132-4, 721-4826, and -70145. Here 

 the numbers are written under each other as in 

 the margin ; the hundreds under 

 hundreds, tens under tens, units ..** 

 under units, tenths under tent '/is, 133-4 

 hundredths under hundredths, thou- 721-4826 

 sandths under thousandths, &c. ; -70145 

 then the several units of the same 

 order are added as before. Some 859-33405 

 care is here necessary in writing 

 down the several numbers, in order that the 

 units of the same order may be placed 

 under each other ; but we can easily assure 

 ourselves of this by having the decimal points 

 in each of the numbers all arranged under 

 each other. 



Subtraction is that process by which we di- 

 minish a number by a given number : or it is that 

 process by which we find the difference between a 

 greater number and a less. 



It is easy from a given number to subtract 

 another small number. Thus, let it be required 

 to subtract or take away 4 from 9. Here we have 

 only to take away i unit from 9 as often as there 

 are units in 4. We say, i from 9 leaves 8, i from 

 8 leaves 7, i from 7 leaves 6, i from 6 leaves 5 ; 

 and therefore 5 is the number left, or 5 is the 

 difference between 9 and 4. This process would, 

 however, prove very tedious if the number to be 

 subtracted was large, and, as in addition, we 

 resolve the numbers into their component units, 

 and find the difference between each correspond- 

 ing set. Thus : To find the difference between 

 7683 and 4361. Here, having written the 

 numbers under each other as in the 7683 

 margin, we say, I unit from 3 units leaves 4^61 

 2, 6 tens from 8 tens leaves 2 tens, 3 hun- 



9543 

 2768 



dreds from 6 hundreds leaves 3 hundreds, 33 22 

 4 thousands from 7 thousands leaves 3 thou- 

 sands ; and the difference is 3322, or 3 thousands, 3 

 hundreds, 2 tens, and 2 units. Again, to find the 

 difference between 845-424 and 23-113. 

 Writing these numbers as in the margin, 841; -424. 

 we say, 3 thousandths from 4 thou- 23-113 



sandths leaves I thousandth, I hun- 

 dredth from 2 hundredths leaves i hun- 822-311 

 dredth, i tenth from 4 tenths leaves 3 

 tenths, 3 units from 5 units leaves 2 units, 2 tens 

 from 4 tens leaves 2 tens, o hundreds from 8 

 hundreds leaves 8 hundreds ; and the result is 

 822-311, or 8 hundreds, 2 tens, 2 units, 3 tenths, I 

 hundredth, and i thousandth. In this way we 



might subtract any number from any larger 

 number, provided the units in the smaller number 

 are less than the corresponding units in the greater 

 number. If, however, this is not the case, we 

 employ a certain device, which enables us to 

 obtain the required difference between the two 

 given numbers ; and this we now proceed to 

 explain. 



In the first place, it is necessary to see that the 

 difference between two numbers is not altered if 

 we increase each of the given numbers by the same 

 number : thus, the difference between 5 

 and 3 is 2. Now, if to 5, and also to 3, 5 5 9 

 i we add, say I, thus making the numbers ; 4 7 

 6 and 4, the difference is still 2. Again, : 

 if to 5, and also to 3, we add 4, the 222 

 difference between the resulting num- 

 bers, 9 and 7, is still 2 ; that is, the difference 

 remains unaltered. This is evidently what we 

 might expect, for whatever be the difference 

 between the resulting numbers, it is clear that 

 it cannot arise from the equal numbers which 

 have been added, but only from the original 

 1 numbers. 



Now, by applying this principle, we may find 

 the difference between any two numbers whatever. 

 Thus, let it be required to subtract 2768 from 

 i 9543- Writing the less number under the 

 greater, as in the margin, the units of the 

 same order being placed under each other, 

 we find we cannot take 8 units from 3 

 units, nor 6 tens from 4 tens, 7 hundreds 

 from 5 hundreds. Now, to avoid this difficulty, we 

 shall in the first place add to both numbers 10 

 units ; the addition of these 10 units, we have seen, 

 does not alter in any way the difference between 

 the original numbers. We shall add these units 

 to the upper number as 10 units, and 

 to the lower number as i ten ; the 

 number will then stand as in the 

 margin. We can take 8 units from 

 13 units, but we cannot take 7 tens 

 from 4 tens. We shall then add to 

 both numbers 10 tens, and, as before, this does 

 not alter the difference of the given numbers. 

 We shall add these 10 tens to the upper line as 

 tens, adding 10 of them, and to the under line as 

 hundreds, adding i ; the numbers 

 will then stand as in the margin. 

 The units in the upper line are 

 greater than the units below, the 

 tens above are greater than the tens 

 below, but the hundreds above are 

 less than those below, and therefore, to both 

 numbers we add 10 hundreds, or I thousand ; add 

 this to the upper line as hundreds, and to the 

 under as thousands, the numbers 

 will then stand as in the margin. 

 Now, as the numbers below are each 

 less than the corresponding numbers 

 above, the operation may be per- 

 formed : thus, we say, 8 from 13 

 leaves 5, 7 from 14 leaves 7, 8 from 

 15 leaves 7, 3 from 9 leaves 6 ; and the result is 

 6775, or 6 thousands, 7 hundreds, 7 tens, and 

 5 units. Now, it will at once be seen that QCA-I 

 by this process we have not found the 2768 

 difference between the original numbers, 

 but, what is the same thing, we have found 6775 

 the difference between these numbers after 

 each has been increased by one thousand, one 



th. 

 9 



4 13 

 7 8 



th. 

 9 



2 



th. h. t. 



9 IS H 



3 8 7 



6 7 7 



13 





