What Is the Minus Quantity?
Template:WilliamBGreene This essay appeared in The Massachusetts Teacher, over the signature "W. B. G." The authorship is uncertain. Like "Influence", however, which appeared in the same journal in Jan. 1857, it bears enough resemblance to the work of William Batchelder Greene to warrant inclusion here.
- W. B. G, “What Is the Minus Quantity?,” Massachusetts Teacher and Journal of Home and School Education 13, no. 9 (September 1860): 330-333.
- ---. ---. Vermont School Journal and Family Visitor. 3, 10 (October 1861), 292-295.
IN the discussion of this question, "What is the Minus Quantity?" we must first understand what is meant by a quantity. We believe a quantity to be anything which can be measured. The term quantity is, also, sometimes applied to the expression for it. We will use the term as applicable to either the expression or the thing, but mainly to the expression for it; as, 2, 4, a, or b.
Calling a the expression for a quantity, we wish to learn the difference between a and —a. There evidently is a difference of some kind, or there would be no utility whatever in having a minus quantity. The only perceptible difference seems to be, that, in the latter quantity, a is preceded by the sign minus. And so we infer, from this outward expression, that a minus quantity is a quantity with a minus sign preceding it; and so it is; but this definition does not develop at all the nature or value of the quantity.
To know this, we must have recourse to the method by which it was obtained. Before this, Let us notice some of the current explanations of the minus quantity. It is understood by some to be less than no quantity. I have known pupils, who have made commendable progress in algebra, to state it as their belief, that the minus quantity denoted an absence of quantity, or was less than no quantity, or, to use common language, was less than nothing. The fallacy of this opinion is evident, for what is not a quantity certainly cannot be a minus quantity
Again, it is said that a minus quantity is just like a plus quantity, with the exception that the minus quantity shows a different relation. Now, what is meant by showing a different relation? It is usually explained by drawing a horizontal line, then taking some point near the centre, as a starting—point, and saying that quantity is reckoned in two directions, and that it increases in both directions equally, the only difference being this, that one is to be added and the other subtracted. Do we not have quantities to be added and subtracted in arithmetic? Then the difference would be, the sign shows what is to be done in algebra; but in arithmetic, we are told by word of mouth, or by the printed words of the book. So we should find no real difference between algebra and arithmetic It seems that the ideas gained from an explanation like this are exceedingly vague and indefinite, and tend more to confuse than to make clear the real difference between a plus and minus quantity.
Let us now look at the process by which it is obtained. Take the quantity a, which is an indefinite, known quantity, (by an indefinite, known quantity I mean a quantity for which any definite, known quantity, as 2, 4, or 6, can be substituted); from this, subtract the indefinite known quantity b, and it gives the formula a—b, which is an expression for the difference of any two numbers.
Now, substitute for a the number 3, and for b, 1, the expression will then be 3—1, which reduced =2; substituting for b the number 2, the expression will read 3—2=1. Again, substituting for b the number 3, it gives 3—3=0. It will be noticed that the larger the number that is subtracted the less the result will be.
If we again substitute for b the number 4, for we have as much right to substitute 4 as any number, since b is indefinite, the expression will read 3—4 reduced =—1. Substituting 5 for b the reduced expression will be —2. Applying the rule, which we have found true, that is, that the minuend remaining the same, the greater the subtrahend the less the result; then the result —1 must be considered less than 0, and the result —2 less than either O or —1. Hence it follows that, as —1 is less than 0, that +1 and —1 are unequal, that is, that +1 is greater than —1; and as +1 is the result of adding 1 to 0, and —1 is the result of subtracting 1 from 0, it follows that +1 is greater than —1, by 2.
Again, of two minus quantities, it follows that that is the greater which is the result of the subtraction of the less number; for instance, 3—4=—1, and 3—5=—2. In the first case, 4 is subtracted, in the second case, 5; therefore —1 > —2 because it is the result of the subtraction of the less number, likewise —4 > —6. From this illustration it will be seen—
1. That a minus quantity arises from subtraction, and that it is an expression for the difference which arises from subtracting a greater quantity from a less.
2. That there is an inequality existing between a plus and minus quantity represented by the same symbol,
3. That of two minus quantities that is an expression for the greater which is numerically the less.
It may, perhaps, be seen more clearly by a few illustrations which show the practical working of it when considered in this light. It is evident that +1 is not equal to —1, for if that is true, then the unreduced expression for +1, which is 3—2, must be equal to the unreduced expression for —1, which is 3—4; or to another unreduced expression, 2—3, which is absurd. And to say that +1 = —1, is the same as to say that taking 2 from 3 is the same as taking 3 from 2.
Again, if the same quantities be added at different times, the results will be the same or equal; and, conversely, if different quantities be added, the results will be different or unequal; and if the results are unequal, it follows that the quantities that were added were unequal. Then, taking the quantity 2, to that add 1, and the result will be 3; to it add—1, and the result will be 1. This last result is less by 2 than the first; consequently, the numbers added are unequal. But 2 = 2, therefore +1 is greater by 2 than—1, because the result is greater by 2.
I have heretofore confined the illustrations to definite, known quantities I will give one more demonstration by using indefinite known quantities, which will prove the proposition true for any value of the minus quantity. Now, suppose we admit for the time that 0 <—a, and that—a <—(a + m); then, if we add a + m to both members of the first inequality, 0 <—a, we shall have a + m < m; which is absurd. Adding the same to the inequality, —a <—(a + m), we shall have m < 0; which is absurd. On the contrary, if we take the inequalities 0 >—a and —a >—(a + m), which we have shown to be true, with definite, known numbers, and to them add the same quantities, a + m, we shall have a + m > m and m > 0; which is true.
Knowing that this view of the question gives us a key to many of the intricate operations in the science of algebra, I submit the question to teachers, Would it not facilitate the progress of our pupils in this interesting study to explain, early in their course, the nature of the minus quantity?
W. B. G. *nbsp;