Art. 79. ADDITION is bringing quantities together, to find their amount. On the contrary, SUBTRACTION is FINDING THE DIFFERENCE OF TWO QUANTITIES, OR SETS OP QUANTITIES.
Particular rules might be given, for the several cases in subtraction. But it is more convenient to have one general rule, founded on the principle, that taking away a positive quantity, from an algebraic expression, is the same in effect, as annexing an equal negative quantity; and taking away a negative quantity is the same, as annexing an equal positive one.
Suppose +b is to be subtracted from a + b
Taking away +b from a + b, leaves a
And annexing -b, to a + b, gives a + b - b
But by axiom 5th, a + b - b is equal to a
That is, taking away a positive term, from an algebraic expression, is the same in effect, as annexing an equal negative term.
Again, suppose -b is to be subtracted from a - b
Taking away -b, from a - b, leaves a
And annexing +b to a - b, gives a -b + b
But a - b + b is equal to a
That is, taking away a negative term, is equivalent to annexing a positive one. If an estate is encumbered with a debt; to cancel this debt is to add so much to the value of the estate. Subtracting an item from one side of a book account, will produce the same alteration in the balance, as adding an equal sum to the opposite side.
To place this in another point of view. If m is added to b, the sum is by the notation b + m
But if m is subtracted from b, the remainder is b - m
So if m and h are each added to b, the sum is b + m + h
But if m and h are each subtracted from b, the remainder is b - m - h
The only difference then between adding a positive quantity and subtracting it, is, that the sign is changed from + to -.
Again, if m - n is subtracted from b, the remainder is, b - m + n.
For the less the quantity subtracted, the greater will be the remainder. But in the expression m - n, m is diminished by n; therefore, b - m must be increased by n; so as to become b - m + n: that is, m - n is subtracted from b, by changing +m into -m, and -n into +n, and then writing them after b, as in addition. The explanation will be the same, if there are several quantities which have the negative sign. Hence,
80. TO PERFORM SUBTRACTION IN ALGEBRA, CHANGE THE SIGNS OP ALL THE QUANTITIES TO BE SUBTRACTED, OR SUPPOSE THEM TO BE CHANGED, FROM + TO -, OR FROM - TO +, AND THEN PROCEED AS IN ADDITION.
The signs are to be changed, in the subtrahend only. Those in the minuend are not to be altered. Although the rule here given is adapted to every case of subtraction; yet there may be an advantage in giving some of the examples in distinct classes.
81. In the first place, the signs, may be alike, and the minuend greater than the subtrahend.
Here, in the first example, the + before 16 is supposed to be changed into -, and then, the signs being unlike, the two terms are brought into one, by the second case of reduction in addition, (Art. 74.) The two next examples are subtracted in the same way. In the three last, the - in the subtrahend, is supposed to be changed into +. It may be well for the learner, at fiist, to write out the examples; and actually to change the signs, instead of merely conceiving them to be changed. When he has become familiar with the operation, he can save himself the trouble of transcribing.
This case is the same as subtraction in arithmetic. The two next cases do not occur in common arithmetic.
82. In the second place, the signs may be alike, and the minuend less than the subtrahend.
The same quantities are given here, as in the preceding* article, for the purpose of comparing them together. But the minuend and subtrahend are made to change places. The mode of subtracting is the same. In this class, a greater quantity is taken from a less: in the preceding, a less from a greater. By comparing them, it will be seen, that there is no difference in the answers, except that the signs are opposite. Thus 16b - 12b is the same as 12b - 16b, except that one is +4b, and the other -4b; That is, a greater quantity subtracted from a less, gives the same result, as a less subtracted from a greater, except that the one is positive, and the other negative. See Art. 58 and 59.
83. In the third place, the signs may be unlike.
From these examples, it will be seen that the difference between a positive and a negative quantity, may be greater than either of the two quantities. In a thermometer, the difference between 28 degrees above cypher, and 16 below, is 44 degrees. The difference between gaining 1000 dollars in trade, and losing 500, is equivalent to 1500 dollars.
84. Subtraction may be proved, as in arithmetic, by adding the remainder to the subtrahend. The sum ought to be equal to the minuend, upon the obvious principle, that the difference of two quantities added to one of them, is equal to the other This serves not only to correct any particular error, but to verify the general rule.
85. When there are several terms alike, they may be reduced as in addition.
1. From ab subtract 3am + am + 7am + 2am + 6am.
Ans. ab - 3am - am - 7am - 2am - 6am = ab - 19am. (Art. 72.)
2. From ax - bc + 3ax + 7bc, subtract 4bc - 2ax + bc + 4ax.
Ans. ax - bc + 3ax + 7bc - 4bc + 2ax - bc - 4ax = 2ax + bc. (Art. 78.)
86. When the letters in the minuend are different from those in the subtrahend, the latter are subtracted, by first changing the signs, and then placi?ng the several terms one after another, as in addition. (Art, 79.)
From 3ab + 8 - my + dh subtract x - dr + 4hy - bmx.
Ans. 3ab + 8 - my + dh - x + dr - 4hy + bmx.
Thus a - (b - c+ d) signifies that the quantities b, -c, and +d are to be subtracted from a. The expression will then become a - b + c - d.
2. 7abc - 8 + 7x - (3abc - 8 - dx + r) = 4abc + 7x + dx - r.
3. 6am - dy + 8 - (16 + 3dy - 8 + am - e + r) = ?
86 c. On the other hand, when a number of quantities are introduced within the marks of parenthesis, with - immediately preceding; the signs must be changed.
Thus -m + b - dx + 3h = -(m - b + dx - 3h.)