18-6=3 21-7-3 24÷8=3 27-9=3 36-9-4 23. Division may also be represented by placing the divisor under the dividend, with a short horizontal line be tween them; thus, denotes that 10 is to be divided by 2. In the same way we have 12=12÷2; 1=13÷3; 2=17÷5; 53=53÷7. This method is employed, when in division there is a remainder, to express accurately the value of the quotient. What does division teach? What is the number to be divided called? What is the number by which we divide called? What is the number of times which the dividend contains the divisor called? There is sometimes another part, what is it? Of what name is the remainder? What is the symbol of division? By what other method is division denoted? When the divisor consists of only one figure, we proceed as follows: Divide 973 by 7. Having placed the divisor at the left of the dividend, keeping them separate by means of a curved line, we draw a straight horizontal line underneath. OPERATION. 7)973 139 quotient. We then say, 7 is contained in 9, 1 time and 2 remainder; we write the 1 underneath. As the 9 occupies the hundreds' place, the 2 remainder must be 2 hundreds. The next figure, 7, to be divided, is tens, to which we add the 2 hundreds, or 20 tens, making 27 tens; which result is obtained by prefixing the 2 to the 7. Next, we see how many times 7 is contained in 27, which is 3 times and 6 remainder; we place the 3 for the next figure of the quotient, and conceive the 6 to be prefixed to the next figure of the dividend, making 63; which is the same as adding 6 tens or 60 units to the 3 units. Finally, we find 7 is contained in 63, 9 times. Thus 7 is contained 139 times in 973. Hence, 139 repeated 7 times must equal 973. 24. Suppose we wish to know how many times 8 is contained in 32. We might proceed as follows: since 32 is greater than 8, we know that 8 is contained in it, at least once; therefore, subtracting 8 from 32, we find 24 for a remainder. Again, we know that 8 is contained at least once in 24; therefore, subtracting 8 from 24, we have 16, from which, subtracting 8, we have left 8; finally, from 8 subtracting 8, we have no remainder. Hence, we perceive that 8 has been subtracted 4 times from 32, that is, 8 is contained just four times in 32. It is obvious that by continued subtractions any operation in division may be performed. For this reason division is said to be a concise way of performing several subtractions. CASE I. 25. Short Division is the method of operation wher the divisor consists of only one figure. From the preceding operation we infer the following RULE. 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 O 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. 27. Long Division is the method of operation when the divisor consists of more than one figure. EXAMPLES. 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 OPERATION. DIVISOR. DIVIDEND. QUOTIENT. 354) 4703598 (13287 1062 second product. 1015 708 third product. 3079 2832 fourth product. 2478 2478 fifth product. from 470, we have 116 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 RULE. 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 annex the next 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 annex the next figure of the dividend. Thus continue the operation until all the figures 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. |