ARITHMETIC. 



684-234 

 247653 

 659-4687 



hundred, and ten. This explains what is usually 

 called the process of borrowing and carrying. 



Again, let it be required to subtract 24-7653 

 from 684-234. 



Writing the less number under the greater, as 

 in the margin, we begin by adding to 

 both 10 tenths of thousandths ; we add 

 this to the upper number as tenths of 

 thousandths, and to the other as I 

 thousandth. Again we add to both 10 

 thousandths, adding it as such to the 

 upper number, and to the under as i hundredth. 

 Again we add to both 10 hundredth*, adding it as 

 such to the upper number, and to the under as I 

 tenth. Again adding to both 10 tenths, adding it as 

 such to the upper number, and to the under as I 

 unit ; and so on. This device will now enable us to 

 perform the subtraction, for we say, 3 from 10 

 there remains 7, 6 from 14 there remains 8, 7 

 from 13 there remains 6, 8 from 12 there remains 



4, 5 from 14 there remains 9, 3 from 8 there is 



5, and o from 6 remains 6. In this way any whole 

 or decimal number can be subtracted from any 

 other number which is greater. 



MULTIPLICATION OF WHOLE AND DECIMAL 

 NUMBERS. 



Multiplication is that process by which, having 

 given two numbers, we find a third, composed of 

 either of the given numbers, in the same manner as 

 the other is composed of the unit. Thus, for in- 

 stance, if we are to multiply 7 by 4 ; this means 

 that we are to perform on 7 the same operation as 

 has been performed on the unit to produce 4. 

 Now, to produce 4, the unit has evidently been 

 repeated 4 times, hence we must also repeat 7 four 

 times, as in the margin ; the resulting number, 

 28, is said to be the product of 7 by 4. In the 7 

 same manner we might multiply any number 7 

 by any other number ; as, for instance, 7459 7 

 by 365. Here we would require to write the 7 

 number 7459 under itself 365 times, and add ; 

 this would involve an immense amount of 28 

 trouble, which we can save ourselves. In the 

 first place, the products of all the numbers up to 

 9 by all the numbers up to 9 have been 

 found and tabulated in what is called the mul- 7459 

 tiplication table, and these results serve us 7459 

 for obtaining the product of any two num- 7459 

 bers, however large. For example, let it be 

 required to multiply 7459 by 365. Here it 

 is evident that if we write down the number 

 7459 under itself 365 times, the whole may 

 be supposed to consist of three groups : the 

 first group containing 7459 5 times, the second 

 group containing it 60 times, and the 

 third group containing it 300 times. 7459^ 

 Now the sum of the numbers in the first 7459 I o. 

 group, which contains 7459 5 times, may 7459 1 |- 

 be found as here shewn ; but it is clear 7459 I 8 

 that instead of taking the sum of 5 nines, 745gJ 

 5 fives, 5 fours, 5 sevens, the multiplica- 7459^ 

 tion table will enable us at once to write 7459 

 the product, for it shews us that 5 times 9 . . . I 

 are 45, 5 times 5 are 25, with 4 carried ... j 

 make 29, &c. The product of 7459 by 

 5 is therefore 37295. Now the product ... j 

 of 7459 by 60 is the same as 10 times the 

 product by 6 ; but the product by 6 is, as in the 

 margin, 44754, and 10 times this is found 



by annexing a zero on the right Again 

 the product of 7459 by 300 is the same as 

 100 times the product by 3 ; but this pro- 

 duct by 3 is 22377, and 100 times this is 

 found by annexing two zeros on the right. 

 It appears, then, that the sum of 7459 

 written under itself 365 times, is the same 

 as the sum of three numbers, viz., the pro- 

 duct of 7459 by 5, the product of 7459 by 

 60, and the product by 300 ; these are as 

 in the margin. These three numbers are 

 now summed, and the product required is 

 found. This process may be abridged as is here 



7459 

 7459 

 7459 

 7459 

 7459 



37295 



7459 

 5 



37295 



7459 

 60 



447540 



7459 

 300 



2237700 



37295 



447540 



2237700 



2722535 



shewn, by writing down 7459 once for all, then as 

 the second product has always one zero on the 

 right, it is not written down, nor are the two zeros 

 on the right of the third product 



7459 

 365 



37295 

 44754 

 22377 



2722535 



604076 

 30104 



2416304 

 604076 

 1812228 



18185103904 



7459 

 365 



37295 

 44754 

 22377 



2722535 



Again, we obtain the product of 604076 by 30104 

 as above. 



In the_same manner may be found the product 

 of any two whole numbers whatever. The num- 

 ber to be multiplied or repeated is called the 

 multiplicand i and that which marks the number 

 of repetitions, the multiplier ; the result is termed 

 the product. Let it now be required to multiply 

 together two decimal numbers ; for example, let it 

 be required to multiply 7-459 by 3-65. Here we 

 will begin by multiplying the whole 

 number 7459 by the whole number 365, 

 and from this result we will deduce the 

 result required. But the product of 

 7459 by 365 is, as in the margin, 2722535. 

 Now, in taking 74 59 for the multiplicand 

 instead of 7-459, we have been making 

 the multiplicand one thousand times too 

 much, the product is therefore one 

 thousand times too much. Again, since 

 we have made the multiplier one hundred times 

 too much, the product is on this account one 

 hundred times too much, so that we must first make 

 the product which we have found one thou- 

 sand times less, and then one hundred times less 

 still. But the product is made one thousand times 

 less by removing the point (which is understood 

 to be on the right of the unit figure of the product) 

 three places to the left ; again the resulting number 

 is made one hundred times less by removing the 

 point two places to the left ; we will, therefore, 

 have for the product of the given numbers 

 27-22535. 



Hence it appears that to multiply together two 

 decimal numbers, we multiply as if they were 

 whole numbers, and cut off as many decimals from 

 the product as there are decimals in the multi- 

 plicand and multiplier together. 



597 



