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1. Place the divisor at the left of the dividend, keeping them separate by a curved line, and draw a straight line underneath the dividend.

II. Seek how many times the divisor is contained in the left-hand figure or figures of the dividend, and place the result directly beneath, for the first figure of the quotient.

III. If there is no remainder, divide the next figure of the dividend for the next figure of the quotient. But when there is a remainder, conceive it to be prefixed to the next succeeding figure of the dividend before making the next division. If a figure of the dividend, which is required to be divided, is less than the divisor, we must write 0 in the quotient, and consider that figure as a remainder.

Division is said to be a concise way of performing what? What is Short Division? Repeat the rule.

EXAMPLES.

1. Divide 2345675 by 8.

OPERATION.

Divisor 8)2345675 dividend.
Quotient 293209 with 3 remainder.

26. When there is a remainder, we may place it over the divisor, with a short horizontal line between them, thus indicating that this remainder is still to be divided by the divisor, agreeably to ART. 23.

2. Divide 12456789 by 4. 3. Divide 78900346 by 7. 4. Divide 131305678 by 6. 5. Divide 357020348 by 3.

Ans. 3114 197+. Ans. 1127 1478. Ans. 218842794. Ans. 119006.7823.

CASE II.

27. Long Division, or when the divisor consists of more

than one figure.

feXAMPLES. 1. Divide 4703598 by 354.

It requires 3 figures, 470, of the dividend to contain the divisor 354. This is contained once in 470; we place the 1 at the right of the dividend for the first figure of the quotient, keeping it separate from the dividend by a curved line. Multiplying the divisor by this quotient figure, and subtracting the product from 470, we have 116

OPERATION.

DIvisor. Dividend. QuotienT.

354) 4703598 (13287
354 first product.

1163
1062 second product.

1015
708 third product.

3079
2832 fourth product.

2478
2478 fifth product.

for a remainder, to which we

annex the next figure, 3, of the dividend, thus forming the number 1163. We now seek how many times the divisor is contained in 1163, which is 3 times. We place the 3 for

a second figure of the quotient.

Multiplying the divisor by this second figure, and subtracting the product from 1163, we find 101 for a second remainder; to which annexing 5, the next figure of the dividend, we have 1015. Thus we proceed till all the figures of the dividend have been brought down. From the above work we readily deduce the following

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I. Place the divisor at the left of the dividend, keeping them separate by a curved line. II. Seek how many times the divisor is contained in the fewest figures of the dividend that will contain it; set the figure expressing the number of times at the right of the dividend for the first figure of the quotient, keeping dividend and quotient separate by means of a curved line. III. Multiply the divisor by this quotient figure, and subtract the product from those figures of the dividend used, and to the remainder annew the newt figure of the dividend; then find how many times the divisor is contained in this new number, and write the result in the quotient. IV. Again, multiply the divisor by this last quotient figure, and subtract the product from the last number which was divided, and to the remainder annew the newt figure of the dividend. Thus continue the operation until all the gures of the dividend have been brought down. Note 1.-Having brought down a new figure, if the number thus formed be less than the divisor, it will contain it 0 times; we therefore write 0 in the quotient, and bring down another figure. NotE 2,-If in multiplying the divisor by any quotient figure we obtain a product which exceeds the number we sought to divide, we must make the quotient figure smaller. Note 3.—If a remainder should be found larger than the divisor, the quotient figure must be taken larger.

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28. If, now, taking the preceding example, we multiply the divisor by the quotient, we shall have this

opertATION.

354 13287

2478 first product. 2832 second product. 708 third product. 1062 fourth product. 354 fifth product.

4703598

Here we discover that the products obtained by this multiplication, are the same as those obtained in the operation of division, only they occur in a reverse order. In the operation of division, each succeeding product is placed one figure farther towards the right, while in the operation of multiplication, each succeeding product is placed one figure farther towards the left. Hence the sum of the products in the case of division, must be the same as the sum in the case of multiplication. In the operation of division, by the above rule, these products are successively subtracted from the corresponding parts of the dividend, until the whole is exhausted. Now we have just shown by the operation of multiplication, that the sum of these products, taken in the order in which they stand, is equal to the dividend. Therefore the above rule for Long Division must be correct,

PROOF.

From what has been said, we also infer that this method of long division proves itself as we proceed with the work, since we have only to add the successive products, and the remainder, if any, to obtain the dividend.

What is Long Division ? How do you piace the numbers 1 Repeat the rule. If, after having brought down a new figure, the result is less than the divisor, how do you proceed? When the partial product is greater than the number which was supposed to contain the divisor, how do you do? When the remainder is greater than the divisor, how do you proceed? Explain the method of proof.

2. Divide 175678 by 223.

OPERATION.

223) 175678 (787
1561 first product.

1957
1784 second product.

1738
1561 third product

177 remainder.

If we take the sum of the successive products and the remainder, adding them as they now stand in the above work, we shall obtain 175678; which, agreeing with the dividend, proves the accuracy of the division. This method of proving division is perhaps as simple and brief as any method which can be devised.

The common method of proving Division, and one which is applicable to Short Division as well as to Long Division, is to multiply the divisor and quotient together, and to add in the remainder, if any.

3. Divide 7892343 by 139. Ans. 56779*. 4. Divide 177575124270 by 753465. Ans. 235678.

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