figure 6 on the left by a unit; by which means (48) we have 599 units of this order instead of 600. 55990999269 (A — B) as before. Let the following examples be performed and proved in like 92. When B is a series of nines, -as the ar. comp. is, in this case always 1,-we have only to diminish A by the alt and add 1 to its unit figure. 1. 4250484 - 999999 = 3250485 Here, as the alt is 6 alt, we merely subtract 1 from the 4 on the left and add 1 to that on the right. 93. A series of nines is the alt minus 1. Therefore, to add a series of nines to a number, is to increase it by the alt and diminish it by a single unit. Thus: 94. The signs and —, symbols of the opposite attributes of quantity, are further distinguished by the following appellatives: The sign+, which shows the increase of quantity, -the operation by which it is numerically formed, and, hence, may be said to affirm its positive or real existence,—is called the affirmative or positive sign: and the sign, the symbol of the operation by which it is diminished and ultimately destroyed, and which may, therefore, be said to deny it existence, is called the negative sign. 95. A number, having on the left of it, is called a positive number, (real number.) A number, having on the left, is called a negative number, (imaginary number;) because, of itself, it expresses a want of the number of units, to render it equal to 0, which as a real number it would express. Thus, 550. Hence, the positive and negative value of the same quantity destroy each other. Numbers without sign are, of course, positive. Note. The expression on the left of the sign = is called the first member, and that on the right, the second member of the equation. = 96. Let a be the positive, and -a the negative value of any quantity or number: also, let d the difference between the two values. Then, if to each value we add a, this (84) will not affect d. Wherefore d is the difference between a+a and a- a: but a a=0, (95;) therefore, d is the diff. between a + a or 2a and 0; consequently, d=2a; that is, the difference between the positive and negative value of a quantity is twice its positive value, or the double of that quantity. Hence, to subtract the negative value of a quantity is the same as to add its positive value. 97. This has sometimes been familiarly illustrated thus: There are two persons, A and B, one of whom, A, has five dollars, (+5;) the other, B, has no money, and is in debt five dollars, (-5.) Now, to discharge his debt-that is, to render him even with the world, or having nothing, (0,) B would require five dollars, and, to be on a par with A, five more therefore, A is 10 dollars better off than B: that is, the diff. between 5 and 5 is 10. Also, as the cancelling or taking away of the debt of B is the same as to give B five dollars, we say, that to subtract 5 is the same as to add 5. Therefore, when several negative quantities are to be subtracted, they must be added as positive quantities—that is, all their signs must be changed from minus to plus. 98. The scholar must not confound the idea of the subtraction of negative quantities with that of their addition. Thus, aa is the addition of the negative value of a to its positive value; or, which is the same thing, the subtraction from a of its positive value. The expression of the subtraction of its negative value, without changing its sign, would stand thus: a- (a,) which looks awkward, and is much better written a + a. Wherefore, (as negatives, or quantity destroyers,) the writing of negative quantities, in any expression, with their appropriate sign-, implies their addition; with a contrary sign, their subtraction. 99. To find the value of an expression in which there are several positive and several negative numbers, add all the positive numbers together: then add all the negative numbers together, the difference between the two sums, preceded by the sign of the greater, is the value of the expression. Then 605 456 =149. The greater sum being positive, the remainder 149, which is the value of the expression, is positive, and is placed on the right of the sign Or, as it is of no consequence in what order the numbers are placed, we may proceed thus: 569+19-34+17-413-9-569+19+17-34-413-9 605-456-149 In this case the work may be performed without placing the numbers under each other. 2. 999-2014 † 164 — 13 — 79 +40: = -903 2106 Sum of neg. num. 1203 Sum of pos. num. 1203 - 2106 903 diff. or required val. In this example, the sum of the negative numbers being the greater, the value of the expression is negative. 5. 1050 5198 +23 14+7 · 120 = = Ascertain IV. The ar. comp. may here be of great use. by the eye whether the result is likely to be positive or negative, writing the ar. comps. of the numbers of contrary sign instead of the number: also taking care (90) to diminish by the alt unit for every complement used. Here, as we easily see that the result will be negative, we write the ar. comps. of the two positive numbers, the alts of which being 4th alt, we diminish the largest negative number by 2 units of this order. The scholar may prove the work by pursuing the ordinary method: 2. 532946-171 — 3436 + 546 — 27 521846 829 ar. comp. of 171 6564 ar. comp. of 3436 546 73 ar. comp. of 27 529858 = 529858 The result being evidently positive, we take the ar. comp. of the three negative numbers: and, as the alts are 2d, 3d, and 4th, we diminish the largest positive number by 1 unit of each of these orders. The scholar may prove as before. = = 3. 54927-9999 + 99 — 9 = - 10. 854327 3745+ 9254 473 49: 11. 747599 274 + 9999 + 999 — 8451 = = 747597 +726 + 1549 749872 (90 and 93) 12 — 43560 — 99999999+ 547 +57 +9999 = SECTION V. MULTIPLICATION. 101. The word multiplication is derived from two Latin words-multus, many, and plicare, to fold. As an operation, it is the method of finding, with greater facility than by either Numeration or Addition, the sum produced by the combination of all the units expressed by writing a number two or more times. (See Art. 59.) 102. The multiplication of numbers, however great, requires nothing more than the multiplication of a single figure by a single figure. Hence, the following Table, which contains all the products arising from the multiplication of any two of the figures 2, 3, 4, 5, 6, 7, 8, 9, will prepare the scholar to perform any multiplication whatever. To make the exercise complete, he must perform each multiplication, where the factors are not alike, both ways. Thus, for the expression 2 × 3 = 6, he must say, twice 3 is 6; three times 2 is 6. For the expression 2 × 48, he must say, twice 4 is 8; four times 2 is 8, and so on. 103. The numbers produced by the continual addition of each of the digits form the following Table, which is attributed to Pythagoras. The first line is formed by adding 1 to itself, and continuing the addition till we have 9 times 1. The second line, by adding 2 in the same manner. The third, by adding 3, and so on for the others. |