Addition

Art. 64. In entering on an algebraic calculation, the first thing to be done, is evidently to collect the materials. Several distinct quantities are to be concerned in the process. These must be brought together. They must be connected in some form of expression, which will present them at once to our view, and show the relations which they have to each other. This collecting-of quantities is what, in algebra, is called addition. It may be defined, THE CONNECTING OF SEVERAL QUANTITIES, WITH THEIR SIGNS, IN ONE ALGEBRAIC EXPRESSION.

65. It is common to include in the definition, "uniting in one term, such quantities, as will admit of being united." But this is not so much a part of the addition itself, as a reduction, which accompanies or follows it. The addition may, in all cases be performed, by merely connecting the quantities by their proper signs. Thus a added to b, is evidently a and b: that is, according to the algebraic notation, a + b. And a added to the sum of b and c, is a + b + c. And a + b, added to c + d, is a + b + c + d. In the same manner, if the sum of any quantities whatever, be added to the sum of any others, the expression for the whole, will contain all these quantities connected by the sign +.

66. Again, if the difference of a and b be added to c; the sum will be a - b added to c, that is a - b + c. And if a - b be added to c - d, the sum will be a - b + c - d. In one of the compound quantities added here, a is to be diminished by b, and in the other, c is to be diminished by d; the sum of a and c must therefore be diminished, both by b, and by d, that is, the expression for the sum total, must contain -b and -d. On the same principle, all the quantities which, in the parts to be added, have the negative sign, must retain this sign in the amount. Thus a + 2b - c, added to d - h - m, is a + 2b - c + d - h - m.

67. The sign must be retained also, when a positive quantity is to be added, to a single negative quantity. If a be added to -b, the sum will be -b + a. Here it may be objected, that the negative sign prefixed to b, shows that it is to be subtracted. What propriety then can there be in adding it? In reply to this, it may be observed, that the sign prefixed to b while standing alone, signifies, that b is to be subtracted, not from a, but from some other quantity, which is not here expressed. Thus -b may represent the loss, which is to be subtracted from the stock in trade. (Art. 55.) The object of the calculation, however, may not require that the value of this stock should be specified. But the loss is to be connected with a profit on some other article. Suppose the profit is 2000 dollars, and the loss 400. The inquiry then, is what is the value of 2000 dollars profit, when connected with 400 dollars loss?

The answer is evidently 2000-400, which shows that 2000 dollars are to be added to the stock, and 400 subtracted from it; or which will amount to the same, that the difference between 2000 and 400 is to be added to the stock.

68. QUANTITIES ARE ADDED, then, BY WRITING THEM ONE AFTER ANOTHER, WITHOUT ALTERING THEIR SIGNS; observing always, that a quantity, to which no sign is prefixed, is to be considered positive. (Art. 29.)

The sum of a + m, and b - 8, and 2h - 3m + d, and h - n and r + 3m - y, is
     a + m + b - 8 + 2h - 3m + d + h - n + r + 3m - y.

69. It is immaterial in what order the terms are arranged. The sum of a and b and c is either a + b + c, or a + c + b, or c + b + a. For it evidently makes no difference, which of the quantities is added first. The sum of 6 and 3 and 9, is the same as 3 and 9 and 6, or 9 and 6 and 3.

And a+ m - n, is the same as a - n + m. For it is plainly of no consequence, whether we first add m to a, and afterwards subtract n; or first subtract n and then add m.

70. Though connecting quantities by their signs is all which is essential to addition; yet it is desirable to make the expression as simple as may be, by reducing several terms to one. The amount of 3a, and 6b, and 4a, and 5b, is
         3a + 6b + 4a + 5b.
But this may be abridged. The first and third terms may he brought into one; and so may the second and fourth. For 3 times a, and 4 times a, make 7 times a. And 6 times b, and 5 times b, make 11 times b. The sum when reduced is therefore 7a + 11b.

For making the reductions connected with addition, two rules are given, adapted to the two cases, in one of which, the quantities and signs are alike, and in the other, the quantities are alike, but the signs are unlike. Like quantities are the same powers of the same letters. (Art. 45.) But as the addition of powers and radical quantities will be considered in a future section, the examples given in this place, will be all of the first power.

71. Case I. TO REDUCE SEVERAL TERMS TO ONE, WHEN THE QUANTITIES ARE ALIKE, AND THE SIGNS ALIKE, ADD THE COEFFICIENTS, ANNEX THE COMMON LETTER OR LETTERS, AND PREFIX THE COMMON SIGN.

Thus to reduce 3b + 7b, that is +3b + 7b to one term, add the coefficients 3 and 7; to the sum 10, annex the common letter b, and prefix the sign +. The expression will then be +10b. That 3 times any quantity, ana 7 times the same -quantity, make 10 times that quantity, needs no proof.

Examples.

bc + 2bc + 9bc + 3bc = 15bc

7b + xy + 8b + 3xy + 2b + 2xy + 6b + 5xy = 23b + 11xy

cdxy + 3mg + 2cdxy + mg + 5cdxy + 7mg + 7cdxy + 8mg = 15cdxy + 19mg

The mode of proceeding will be the same, if the signs are negative.

Thus -3bc - bc - 5bc, becomes, when reduced, - 9bc.

And -ax - 3ax - 2ax = -6ax. Or thus,

-3bc - bc - 5bc = 9bc

-2ab - my - ab - 3my - 7ab - 8my = -10ab - 12my

72. It may perhaps be asked here, as in art. 68, what propriety there is, in adding quantities, to which the negative sign is prefixed; a sign which denotes subtraction? The answer to this is, that when the negative sign is applied to several quantities, it is intended to indicate that th^se quantities are to be subtracted, not from each other, but from some other quantity marked with the contrary sign. Suppose that, in estimating a man's property, the siun of money in his possession is marked +, and the debts which he owes are marked -. If these debts are 200, 300, 500 and 700 dollars, and if a is put for 100; they will together be-2a - 3a - 5a - 7a. And the several terms reduced to one, will evidently be -17a, that is, 1700 dollars.

73. Case II. TO REDUCE SEVERAL TERMS TO ONE, WHEN THE QUANTITIES ARE ALIKE, BUT THE SIGNS UNLIKE, TAKE THE LESS COEFFICIENT FROM THE GREATER; TO THE DIFFERENCE, ANNEX THE COMMON LETTER OR LETTERS, ND PREFIX THE SIGN OF THE GREATER COEFFICIENT.

Thus, instead of 8a - 6a, we may write 2a.

And instead of 7b - 2b, we may put 5b.

74. Here again, it may excite surprise, that what appears to be subtraction, should be introduced under addition. But according to what has been observed, (Art. 66.) this subtraction is strictly speaking, no part of the addition. It belongs to a consequent reduction. Suppose 6b is to be added to a - 4b. The sum is a - 4b + 6b. (Art. 69.)

But this expression may be rendered more simple. As it now stands, 4b is to be subtracted from a, and 6b added. But the amount will be the same, if, without subtracting any thing, we add 2b, making the whole a + 2b. And in all similar instances, the balance of two or more quantities, ma} be substituted for the quantities themselves.

75. If two equal quantities have contrary signs, they destroy each other, and may be cancelled. Thus+6b- 6b =0. And 3.6 - 18 = 0. And 7bc - 7bc = 0.

Let there be any two quantities whatever, of which a is the greater, and b the less. Their sum will be a + b. And their difference a - b. The sum and difference added, will be 2a + 0, or simply 2a. That is, if the sum and difference of any two quantities be added together, the whole will be twice the greater quantity. This is one instance, among multitudes, of the rapidity with which general truths are discovered and demonstrated in algebra. (Art. 23.)

76. If several positive, and several negative quantities are to be reduced to one term ; first reduce those which are positive, next those which are negative, and then take the difference of the coefficients, of the two terms thus found.

Ex. 1. Reduce 13b + 6b + b - 4b - 6b -7b, to one term. By art. 72, 13b + 6b + b = 20b. And -4b - 5b - 7b = -16b. By art. 74, 20b - 16b = 4b, which is the value of all the given quantities, taken together.

Ex. 2. Reduce 3xy - xy + 2xy - 7xy + 4xy - 9xy + 7xy - 6xy. The positive terms are 3xy + 2xy + 4xy + 7xy = 16xy. The negative terms are -xy - 7xy - 9xy - 6xy = -23xy. Then 16xy - 23xy = -7xy.

Ex. 3. 3ad - 6ad + ad + 7ad - 2ad + 9ad - 8ad - 4ad = 0.

4. 2abm - abm + 7abm - 3abm + 7abm = ?

77. If the letters, in the several terms to be added, are different, they can only be placed after each other, with their proper signs. They cannot be united in one simple term. If 4b, and -6y, and 3x, and 17h, and -5d, and 6, be added; " their sum will be
         4b - 6y + 3x + 17h - 5d + 6. (Art. 69.)

Different letters can no more be united in the same term, than dollars and guineas can be added, so as to make a single sum. Six guineas and 4 dollars are neither ten guineas nor ten dollars. Seven hundred and five dozen, are neither 12 hundred nor 12 dczen. But, in such cases, the algebraic signs serve to show how the different quantities stand related to each other; and to indicate future operations, which are to be performed, whenever the letters are converted into numbers. In the expression a-f-6* the two terms cannot be united in one. But if a stands for 15, and if, in the course of a calculation, this number is restored; then a-{-6 will become 15+6, which k equivalent to the single term 21. In the same manner, a - 6, becomes 15-6, which is equal to 9. The signs keep in view the relations of the quantities till an opportunity occurs of reducing several terms to one.

78. When the quantities to be added contain several terms which are alike, and several which are unlike, it will be convenient to arrange them in such a manner, that the similar terms may stand one under another.

Examples.

1. Add and reduce ab + 8 to cd - 3 and 5ab - 4m + 2. The sum is 6ab - 7 + cd - 4m.

2. Add x + 3y - dx, to 7 - x - 8 + hm.
Ans. 3y - dx - 1 + hm.

3. Add abm - 3x + bm, to y - x + 7 and 5x - 6y + 9.

4. Add 3am + 6 - 7xy - 8, to 10xy - 9 + 5am.

5. Add 6ahy + 7d - 1 + mxy, to 3ahy - 7d + 17 - mxy.

6. Add 2by - 3ax + 2a, to 3bx - by + a.

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