Positive and Negative Quantities
53. To one who has just entered on the study of algebra, there is generally nothing more perplexing, than the use of what are called negative quantities. He supposes he is about to be introduced to a class of quantities which are entirely new; a sort of mathematical nothings, of which he can form no distinct conception. As positive quantities are real, he concludes that those which are negative must be imaginary. But this is owing to a misapprehension of the term negative, as used in the mathematics.
54. A NEGATIVE quantity is one which is required to be SUBTRACTED. When several quantities enter into a calculation, it is frequently necessary that some of them should be added together, while others are subtracted. The former are called affirmative or positive, and are marked with the sign +; the latter are termed negative, and distinguished by the sign -. If, for instance, the profits of trade are the subject of calculation, and the gain is considered positive; the loss will be negative; because the latter must be subtracted from the former, to determine the clear profit. If the sums of a book account are brought into an algebraic process, the debt and the credit are distinguished by opposite signs. If a man on a journey is, by any accident, necessitated to return several miles, this backward motion is to be considered negative, because that, in determining his real progress, it must be subtracted from the distance which he has travelled in the opposite direction. If the ascent of a body from the earth be called positive, its descent will be negative. These are only different examples of the same general principle. In each of the instances, one of the quantities is-to be subtracted from the other.
55. The terms positive and negative, as used in the mathematics, are merely relative. They imply that there is, eitherin the nature of the quantities, or in their circumstances, or in the purposes which they are to answer in calculation, some such opposition as requires that one should be subtracted from the other. But this opposition is not that of existence and non-existence, nor of one thing greater than nothing, and another less than nothing. For, in many cases, either or the signs may be, indifferently and at pleasure, applied to the very same quantity; that is, the two characters may change places. In determining the progress of a ship, for instance, her easting may be marked +, and her westing -; or the westing may be +, and the easting -. All that is necessary is, that the two signs be prefixed to the quantities, in such a manner as to show, which are to be added, and which subtracted. In different processes, they may be differently applied. On one occasion, a downward motion may be called positive, and on another occasion negative.
56. In every algebraic calculation, some one of the quantities must be fixed upon, to be considered positive. All other quantities which will increase this, must be positive also. But those which will tend to diminish it, must be negative. In a mercantile concern, if the stock is supposed to be positive, the profits will be positive; for they increase the stock; they are to be added to it. But the losses will be negative; for they dimmish the stock; they are to be subtracted from it. When a boat, in attempting to ascend a river, is occasionally driven back by the current; if the progress up the stream, to any particular point, is considered positive, every succeeding instance of forward motion will be positive, while the backward motion will be negative.
57. A negative quantity is frequently greater, than the positive one with which it is connected. But how, it may be asked, can the former be subtracted from the latter? The greater is certainly not contained in the less: how then can it be taken out of it? The answer to this is, that the greater may be supposed first to exhaust the less, and then to leave a remainder equal to the difference between the two. If a man has in his possession 1000 dollars, and has contracted a debt of 1500; the latter subtracted from the former, not only exhausts the whole of it, but leaves a balance of 500 against him. In common language, he is 500 dollars worse than nothing.
58. In this way, it frequently happens, in the course of an algebraic piocess, that a negative quantity is brought to stand alone. It has the sign of subtraction, without being connected with any other quantity, from which it is to be subtracted. This denotes that a previous subtraction has left a remainder, which is a part of the quantity subtracted. If the latitude of a ship which is 20 degrees north of the equator, is considered positive, and if she sails south 25 degrees; her motion first diminishes her latitude, then reduces it to nothing, and finally gives her 5 degrees of south latitude. The sign - prefixed to the 25 degrees, is retained before the 5, to show that this is what remains of the southward motion, after balancing the 20 degrees of north latitude. If the motion southward is only 15 degrees, the remainder must be +5, instead of - 5, to show that it is a part of the ship's northern latitude, which has been thus far diminished, but not reduced to nothing. The balance of a book account will be positive or negative, according as the debt or the credit is the greater of the two. To determine to which side the remainder belongs, the sign must be retained, though there is no other quantity, from which this is again to be subtracted, or to which it is to be added.
59. When a quantity continually decreasing is reduced to nothing, it is sometimes said to become afterwards less than nothing. But this is an exceptionable manner of speaking. No quantity can be really less than nothing. It may be diminished, till it vanishes, and gives place to an opposite quantity. The latitude of a ship crossing the equator, is first made less than nothing, and afterwards contrary to what it was before. The north and south latitudes may therefore oe properly distinguished, by the signs + and -; all the positive degrees being on one side of 0, and all the negative, on the other; thus,
+6, +5, +4, +3, +2, +1, 0, - 1, - 2, - 3, - 4, - 5, &c.
The numbers belonging to any other series of opposite quantities, may be arranged in a similar manner. So that may be conceived to be a kind of dividing point between positive and negative numbers. On a thermometer, the degrees above 0 may be considered positive, and those below 0, negative.
60. A quantity is sometimes said to be subtracted from 0. By this is meant, that it belongs on the negative side of 0. But a quantity is said to be added to 0, when it belongs on the positive side. Thus, in speaking of the degrees of a thermometer, 0 + 6 means 6 degrees above 0; and 0 - 6, 6 degrees below 0.