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we must, in removing the vinculum, change the sign of the number 2 from plus to minus, thus 8 -- 4+2=8-4--2. If from the expresssion 8
(4-2), we remove the vinculum, we have 8—4—2; but this last does not mean that the difference between 4 and 2 is to be taken from 8, but their sum; therefore, as in subtracting 4 we take away 2 too much, we must add 2 to make up the deficiency; consequently, we must, in removing the vinculum, change the sign of the number 2 from minus to plus : thus 8-4-2=8-4+2.
Hence we see that in removing a vinculum preceded by minus, the sign of a positive number under it must be changed from plus to minus, and that of a negative one from minus to plus. Observe, that the sign preceding the vinculum must not be changed.
136. When there are several positive and several negative numbers under a vinculum preceded by minus, as the sum of all the positive numbers is one positive number, and the sum of all the negative, one negative number, the above reasoning applies to the whole of them; consequently, in removing the vinculum, the signs of all the numbers under it must be changed, the positive to negative and the negative to positive, the sign preceding the vinculum remaining the same.
Thus : 967 - (19 +7—8 +3 -- 11) 7=967 - 133 --- 49 + 56
21 + 77 = 1100 — 203 : -897. The student must carefully observe, that when we would remove a vinculum preceded by plus, (98,) the signs of the numbers under it must not be changed. For example : 967 + (19 + 7 - 8 +3 - 11) 7=967 + 133 +49--56
+ 21 — 77 - 1170 — 133 1037. 137. Any two numbers having the sign into between them, are considered as having immediate connection with each other, unless this is otherwise determined by the vinculum. Thus: 6+4 X 3=18 and 4 +6.3=22; but 6 + 4 X 3=30, and (4 + 6)3=30. Again 6--2 X 3=0, but (6 —2)3 =12.
138. From what has been said the student will perceive that when two numbers are to be multiplied, the separation of either or both into parts, at pleasure, has no effect upon the product. For example, 17 X 13 = 221. Now we may, if
we please, separate 17 into the parts 8, 6 and 3; and 13, into the parts 7, 4 and 2, and proceed thus :
product by 7 32 + 24 +12
4 16 + 12 +6
2 56 +42 +21+ 32 +24+ 12+ 16 +12+6 product by 13
The sum of all these products is 221, as before.
Let the following expressions be reduced according to the examples given above : 1. 32-17
+6-14-9=32 2. 186+43 - 27-6 +8+4=184 3. 9645 - (1736 +23) — 17 X 6=778 4. 9645 1736 + (23-17) 6=7945 5. 9645 (1736 + 23 – 17) 6=-807 6. 9645 - 1736 + 23 - 17 X 6=47490
139. DIVISION is the operation by which we find how many times a less number is contained in a greater more easily than by several subtractions. Thus, if we would divide 24 by 6, that is, if we would find how often 6 is contained in 24, instead of operating by several subtractions (60), we recollect (63) how many sixes must be added together to make 24, and say at once 6 is contained 4 times : therefore 4 is the answer.
Also, when a number is given to be divided into a certain number of equal parts, the value of each part is found by division. For if, in the above example, it be required to divide the number 24 into 6 equal parts, we recollect (63) that 6+6+6+6=24; that is to say, that 4 times 6 is 24; and again (59) that this is the same as 6 times 4; that is, 4+4+4+4+4+4=24 : consequently, the number 24 is divided into 6 equal parts, the value of each part being 4.
140. The number from which we subtract, and which is con sequently divided or separated into equal parts, is called the dividend.
The number which we repeatedly subtract, and which naturally signifies the value of each of the equal parts into which the dividend is separated, is called the divisor.
The number which shows how many times the divisor can be subtracted, or how often it is contained in the dividend, and hence the number of equal parts into which, by the operation, the dividend is separated, is called the quotient.
Let us here observe that, where the division is exact, either the divisor or quotient may signify the number of equal parts into which the dividend is separated; the remaining one being the value of each part. Thus, if the divisor signifies the number of parts, the quotient is the value of each part; and, if the divisor signifies the value of each part, the quotient is the number of parts.
141. From what has been said (60 and 139) it is plain that, as the dividend is the product of the divisor and quotient, a division may always be proved by multiplying the divisor and quotient together, which will give the dividend when the work is right.
NOTE. Though this proof is infallible, whatever may be the quotient, the scholar is here only required to make use of it when the divisor is contained an exaet number of times.
142. The following Table will tend to enable the scholar to divide numbers with facility. Let each division in the table (where the divisor and quotient are not alike, that is, where the dividend is not the square of the divisor) be performed both ways: that is, having found the quotient, divide by the quotient, which will give the divisor. Thus, for the expression 6=2 = 3, say 2 in 6 three times; 3 in 6 twice: for the next 8 2 4, say 2 in 8 four times; 4 in 8 twice, and so on throughout the table. Hence, the student may infer that he can always prove an exact division by another division.
DIVISION TABLE. 42 2 9 - 3 3
5 6 2 12 ; 3 4 20; 4 5
30 + 5
: 6 8 15 - 3 5 24 - 4 6
7 10+2=5 18+3=6 28 -4=7
40 = 5
8 12 - 2 -6 21 ; 3=7 32 + 4 = 8 14-2=7 24 -3=8
4 9 36 = 6
6 238 27 : 3
-9 49 + 7 = 7 42 - 6 7 1842 :9
64:8 8 56 : 7 8 48 ; 6 8 81 +9=972+8=9 63 ;-7 91 54 6
143. If a unit or whole thing be divided into two, three, four, five, six, seven, eight, &c. equal parts, the parts are called halves, thirds, fourths, fifths, sixths, sevenths, eighths, &c. In each successive division one part is called one-half, onethird, one-fourth, one-fifth, one-sicth, one-seventh, one-eighth, &c. These parts are expressed in figures thus 1, Ź, , §, ,
1, ], &c.
144. The divisor being the number of equal parts and the quotient the value of each ; if we divide a number by 2, the quotient is one-half that number. If we divide by 3, the quotient is one-third ; and, in general, to divide a number by 4, 5, 6, 7, 8, &c., is to take , 1, , 1, 1, &c. of that number. Also inversely, to take one-half, one-third, one-fourth, &c. of any number, is to divide that number by 2, 3, 4, &c. The expressions i, j, 4, &c. signify 1 divided by 2; 1 divided by 3, &c.
145. As any number or quantity contains itself once, it is plain that a unit divided by a unit is still a unit; also, that a unit is the value of any quantity divided by itself: thus, 2
=1; }=1;=1,&o. As the lower number is always the 2 divisor, it is plain that when this is greater than the upper one, the quotient is less than a unit. In this case the quantity is called a fraction. Thus, which is read four-fifths, 4 divided by 5, or the fifth part of 4, is a fraction. The two numbers which express a fraction are called its terms.
146. As a single unit may be divided or separated into any number of equal parts, at pleasure; any two numbers may
be assumed as dividend and divisor; consequently, the quotient, so far from being always an exact or whole number,
will more frequently be fractional, and sometimes merely a fraction. For example, if we would find how often 9 is contained in 58, we easily perceive (123) that 58 is not an exact multiple, and that the remainder will be 4. Also, as the alt 5 is (123) 5 times 9, plus 5, and, as the 8 together with this 5 exceeds 9 by 4, it is evident that 58 contains 9 six times and 4
express the division of this remainder by 9 thus, g. Wherefore, the operation is read 58 divided by 9 equals 6 and 4 ninths; and written thus, 56 =64.
147. From the preceding we infer that, to find how often 9 is contained in a number expressed by two figures, the sum of which equals or exceeds 9, we have only to add a unit to the left-hand figure. Also that the left-hand figure itself is
the number of times, when the sum of the two is less than 9. Thus 73 =8; and $2 = 3.
148. For more examples, the student may take the numbers that come between those of the division table. Thus, instead of saying 2 in 4 twice, he may say 2 in 5 twice and 1 over; 2 in 7 three times and 1 over. At the next column, instead of saying 3 in 9 three times, he may say 3 in 10 three times and 1 over; 3 in 11 three times and 2 over, and so on for the others. Again, he may say, 23, five by 2 equals two and a half : 1=32, &c.; 10
= 33; 13 16 =51; =53, &c. &c. This will be a valuable exercise.
149. Division presupposes the subtraction of the divisor from the dividend as often as possible : but this subtraction is possible as long as there remains a part of the dividend which equals or exceeds the divisor; and impossible, as soon as this remainder is less than the divisor, even by one unit. Wherefore, the ultimate remainder, when there is one, must be less than the divisor, and may be any number in the natural scale, 1, 2, 3, 4, 5, &c., including the number which is less than the divisor by only one unit. Thus, in dividing by 2, the remainder, if any, is always 1. In dividing by 3, it may be 1 or 2.
In dividing by 4, it may be 1, 2 or 3, and, in like manner, for any other divisor.
150. When a number is given to be separated into a certain number of equal parts, the greater the given number is, the greater will be the value of each part; hence, if we increase the dividend any number of times, leaving the divisor the same, the quotient will be increased the same number of times; that is, (leaving the divisor the same,) to multiply the dividend is to multiply the quotient. Thus, as 6 contains 2 three times, 6 + 6 contains 2 three times plus three times : that is, twice 6 contains 2 twice as often as 6 contains 2, and it is plain that whatever number of times 6 is repeated, the quotient 3 will be repeated the same number of times. Hence, if we divide 6 tens by 2 the quotient will be 3 tens. If we divide 6 hundred by 2, the quotient will be 3 hundred, &c. Now it is evident, that the same reasoning will apply to any two numbers; wherefore, in general, the quotient figure is always of the same order or name as the part divided, which order (48) is determined by that of the right-hand figure of the part divided.
151. As no quotient figure of a superior order can arise