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Ex. 5. Add 3426; 9024; 5106; 8890; 1204, together.
Ans. 27650. 6. Add 509267; 235809; 72920; 8392; 420; 21; and 9, together.
Ans. 826838. 7. Add 2; 19; 817; 4298; 50916 ; 730205; 9180634, together.
Ans. 9966891. 8. How many days are in the twelve calendar months?
Ans. 365. 9. How many days are there from the 15th day of April to the 24th day of November, both days included ? Ans. 224.
10. An army consisting of 52714 infantry*, or foot, 5110 horse, 6250 dragoons, 3927 light-horse, 928 artillery, or gunners, 1410 pioneers, 250 sappers, and 406 miners : what is the whole number of men?
SUBTRACTION teaches to find how much one number exceeds another, called their difference, or the remainder, by taking the less from the greater. The method of doing which is as follows:
Place the less number under the greater, in the same manner as in Addition, that is, units under units, tens under tens, and so on; and draw a line below them.—Begin at the right hand, and take each figure in the lower line, or number, from the figure above it, setting down the remainder below it.But if the figure in the lower line be greater than that above it, first borrow, or add, 10 to the upper one, and then take the lower figure from that sum, setting down the remainder, and carrying 1, for what was borrowed, to the next lower figure, with which proceed as before; and so on till the whole is finished.
* The whole body of foot soldiers is denoted by the word Infantry; and all those that charge on horseback by the word Cavalry. -Some authors conjecture that the term infantry is derived from a certain Infanta of Spain, who, finding that the army
commanded by the king her father had been defeated by the Moors, assembled a body of the people together on foot, with which she engaged and totally routed the enemy. In honour of this event, and to distinguish the foot sol iers, who were not before held in much estimation, they received the name of Infantry, from her own title of Infanta.
TO PROVE SUBTRACTION.
Add the remainder to the less number, or that which is just above it; and if the sum be equal to the greater or uppermost number, the work is right*.
4. From 5331806 take 5073918.
Ans. 257888. 5. From 7020974 take 2766809.
Ans. 4254165. 6. From 8503602 take 574271.
Ans. 7929131. 7. Sir Isaac Newton was born in the year 1642, and he died in 1727: how old was he at the time of his decease?
Ans. 85 years. 8. Homer was born 2543 years ago, and Christ 1810 years ago: then how long before Christ was the birth of Homer?
Ans. 733 years. 9. Noah's flood happened about the year of the world 1656, and the birth of Christ about the year 4000: then how long was the flood before Christ?
Ans. 2344 years. 10. The Arabian or Indian method of notation was first known in England about the year 1150: then how long is it since to this present year 1810?
Ans. 660 years. 11. Gunpowder was invented in the year 1330: then how long was this before the invention of printing, which was in 1441 ?
Ans. 111 years. 12. The mariner's compass was invented in Europe in the year 1302: 'then how long was that before the discovery of America by Columbus, which happened in 1492?
Ans. 190 years.
* The reason of this method of proof is evident; for if the difference of two numbers be added to the less, it must manifestly make up a sum equal to the greater.
MULTIPLICATION is a compendious method of Addition, teaching how to find the amount of any given number when repeated a certain number of times; as, 4 times 6, which is 24.
The number to be multiplied, or repeated, is called the Multiplicand. The number you multiply by, or the number of repetitions, is the Multiplier.–And the number found, being the total amount, is called the Product.-Also, both the multiplier and multiplicand are, in general, named the Terms or Factors.
Before proceeding to any operations in this rule, it is necessary to learn off very perfectly the following Table, of all the products of the first 12 numbers, commonly called the Multiplication Table, or sometimes Pythagoras's Table, from its inventor.
6 9 12 15 18 21 24 27 30 33 36
44 48 5 10 15 20 25 30 35 40 45 50 55 60
8 12 16 20 24 28 32 36 40 441 481
6 12 18 24 30 36 42 | 48
54 601 66 72
7 14 21 28 35 42 49 56 63 70 77 84
8 16 24 32 40 48 56 64 72 80 88 96
9 1827 36 45 54 6372
81 90 99 108
10 20 30 40 50 60 70 80 90 100 110 120
11 22 33 44 55 | 66 77 88 99110 121 132
12 24 | 36 | 48 | 60 / 72 84 196 108/1201132 144
To multiply any Given Number by a Single Figure, or by any
Number not more than 12.
* Set the multiplier under the units figure, or right-hand place, of the multiplicand, and draw a line below it. Then, beginning at the right-hand, multiply every figure in this by the multiplier.-Count how many tens there are in the product of every single figure, and set down the remainder directly under the figure that is multiplied, and if nothing remains, set down a cipher.-Carry as many units or ones as there are tens counted, to the product of the next figures ; and proceed in the same manner till the whole is finished.
Multiply 9876543210 the Multiplicand.
2 the Multiplier.
19753086420 the Product.
To multiply by a Number consisting of Several Figures. + Set the multiplier below the multiplicand, placing them as in Addition, namely, units under units, tens under tens, &c. drawing a line below it.--Multiply the whole of the multiplicand by each figure of the multiplier, as in the last article;
* The reason of this rule is the same as for the process
in Addition, in which 1 is carried for every 10, to the next place, gradually as the several products are produced, one after another, instead of setting them all down one below each other, as in the nexed example.
32 = 8 X 4 280
70 X4 2400 = 600 X 4 20000 5000 X 4
22712 5678 X4
+ After having found the produce of the multiplicand by the first figure of the multiplier, as in the former case, the multiplier is supposed to be divided into parts, and the product is found for the second figure in the same manner : but as this figure stands in the place of tens, the product must be ten times its simple value; and therefore the first figure of this product must be set
in the place of
setting down a line of products for each figure in the multiplier, so as that the first figure of each line may stand straight under the figure multiplying by.--Add all the lines of products together, in the order as they stand, and their sum will be the answer or whole product required.
TO PROVE MULTIPLICATION. THERE are three different ways of proving Multiplication, which are as below:
First Method.—Make the multiplicand and multiplier change places, and multiply the latter by the former in the same manner as before. Then if the product found in this way be the same as the former, the number is right.
Second Method.-* Cast all the 9's out of the sum of the figures in each of the two factors, as in Addition, and set down the remainders. Multiply these two remainders together, and cast the 9's out of the product, as also out of
tens; or, which is the same thing, directly under the figure multiplied by. And proceeding in this manner separately with all the figures of the multiplier, 1234567 the multiplicand. it is evident that we shall mul
4567 tiply all the parts of the multiplicand by all the parts of 8641969 7 times the multe the multiplier, or the whole of 7407402 = 60 times ditto. the multiplicand by the whole 6172835 = 500 times ditto. of the multiplier : therefore 4938268 =4000 times ditto. these several products being added together, will be equal 563826748934567 times ditto. to the whole required product; as in the example annexed.
* This method of proof is derived from the peculiar property of the number 9, mentioned in the proof of Addition, and the reason for the one may serve for that of the other. Another more ample demonstration of this rule may be as follows:-Let P and Q denote the number of g's in the factors to be multiplied, and a and b what remain ; then 9 Pta and 9 Q+b will be the numbers themselves, and their product is (9 P x 9Q) + (9 P X b) + (9 Q x a) + (a x b); but the first three of these products are each a precise number of 9's, because their factors are so, either one or both : these therefore being cast away, there remains only a xb; and if the g's also be cast out of this, the excess is the excess of g's in the total product : but a and b are the excesses in the factors themselves, and a x b is their product; therefore the rule is true.