SUBTRACTION. ] 



MATHEMATICS. ARITHMETIC. 



435 



where 34572 is subtracted from 68594, and the remainde: 

 found to be 34022). 



But if you come to a lower figure, From 6859' 

 which is greater than the figure above Subt. 34572 

 it, add 10 to the upper figure, and then 

 subtract, putting down the remainder as Rem. 34022 

 before, and taking care to carry 1 to the 

 next figure of the lower row. 



For example : let it be required to subtract 27385 

 from 64927 ; then, placing the former 

 number below the latter (as in the mar- From 6492^ 

 gin), we proceed thus : 5 from 7, and 2 Subt. 27385 

 remain : 8 from not 2 but 12, and 4 

 remain ; carry 1 : 4 from 9, and 5 re- Rem. 37642 

 main ; 7 from 14, and 7 remain ; carry 

 1 : 3 from 6, and 3 remain. 



All that requires explanation here is the carrying, as 

 in the former rule. In the preceding example we see 

 that the 8 cannot be taken from the figure above it, 

 because this is only 2 : we, therefore, add 10 to the 2, 

 converting it into 12 ; but the adding 10 to any figure is 

 simply putting 1 before it ; that is, it is adding 1 to the 

 preceding figure, which 1, by carrying it to the next 

 lower or subtractive figure, is taken away again at the 

 next step. In like manner the 4, in the upper row, is 

 converted into 14, and the one thus prefixed to it is 

 afterwards taken away, by 1 being carried to the next 

 lower figure, and 3 subtracted instead of 2. It is plain 

 that in subtraction the carrying can never amount to 

 more than 1. 



As another example, let 86025704 be subtracted from 

 130741392: then, having arranged 

 the numbers as in the margin, we From 130741392 

 proceed thus: 4 from 12, 8; carry Subt 86025704 

 1 : 1 from 9, 8 : 7 from 13, 6 ; carry 



1 : 6 from 11, 5 ; carry 1 : 3 from Rem. 44715688 

 4, 1 : from 7, 7 : 6 from 10, 4 ; 

 carry 1 : 9 from 13, 4 : therefore the remainder is 

 44715688. 



There is a sign far subtraction as well as one for ad 

 ditiou : it is the little mark placed before the number 

 to be subtracted ; it is called minus : 5 2 is therefore 

 5 minus 2, that is, 5 diminished by 2 ; the remainder, 

 or difference, is of course 3 : so that 52 = 3. By help 

 of the plus and minus signs, we can easily connect to- 

 gether in a single row a set of numbers, of which some 

 are to be added, and others to be subtracted ; thus, 

 4+6 3 2 means that 4 and 6 are to be added, and 3 

 and 2 are to be subtracted ; so that 4+6 3 2=5. In- 

 stead of subtracting first 3 and then 2, we may, of 

 course, subtract 5 at once ; so that the above is the 

 same as 10 5=5 ; and whenever addition and subtrac- 

 tion operations are indicated in this way, 

 it will always be best to find first the 125 684 

 sum of the additive quantities, then the 427 !."> 



sum of the subtracting quantities, and 237 



then, as in the foregoing example, to find 699 



the difference of the two results : in this 789 

 manner the result of 125+427 684+ 699 



237 15, is computed as in the margin, 



and found to be 90. So that 125+427 90 result. 

 684+23715=90. 



We shall here add a few examples to be worked in a 

 similar manner : 



1. 861+483 246 179=419. 



1. 573-184+60267=9*4. 



S. 86243+721-649-70+13-9=86249. 



4. 12064+700628 109641 +37 1604=601084. 



5. 23596-624+72311075 137i8-3*062SS+79=SS20S8J. 



In order to prove whether subtraction is correctly per- 

 formed, add the remainder to the number which has 

 been subtracted that is, to the lower of the two pro- 

 posed numbers ; the sum will be the upper number, if 

 the work be correct : thus, in each of the two examples 

 above, we have 



Snbtractive number 27385 86025704 

 Remainder . . . 37642 44715688 



Upper number . 64927 130741302 

 Simple Multiplication. Multiplication is the method 



of finding the mm of any number of equal quantities, 

 without the trouble of repeating them, one under 

 another, and adding them up; it is a short way cJ 

 obtaining the results of addition, when the numbers or 

 quantities to be added are all equal. When the quan- 

 tities are not only equal, but all of one denomination, 

 the operation is called simple multiplication. 



To perform this operation readily, a table, called the 

 multiplication table, must first be learnt ; and the result 

 which arises from multiplying one number by another, 

 provided neither be greater than 12, must be committed 

 to memory; it is one of the few operations in arithmetic 

 where the memory of rules is indispensable. 



The number by which another is to be multiplied, is 

 called the multiplier ; the number which is multiplied, 

 the multiplicand ; and the result obtained which, as 

 just stated, is the same as would be got by writing 

 down the multiplicand as often as there are units in 

 the multiplier, and adding all up, is called the product. 

 The multiplication table shows what the product is in 

 every case in which neither multiplicand nor multiplier 

 exceeds 12 ; and, by knowing this table, the product may 

 always be found, whatever numbers be proposed as mul- 

 tiplicand and multiplier. It may be as well to mention 

 here, that the numbers called by these names, when 

 spoken of together, are generally called factors of the 

 product, as they make or produce it : thus, 2 and 3 are 

 factors of 6, since 3 taken twice, or 2 taken three times, 

 make or produce & 



That the product in any case is really what the table 



itates it to be, the learner can easily prove for him- 



elf ; he has only to take the multiplicand as often as 



;here are units in the multiplier, and, by addition, to 



ind the sum of all ; thus, the table states that 8 times 



are 48, which is true, because 6, written eight times, 



and all added, produce 48, that is, 



and so of any other pair of factors witliin the limits of 

 he table. 



