I found this interesting article about negative numbers. It’s a quote from the paper titled The extension of the natural number domain to the integers in the transition from arithmetic to algebra by Aurora Gallardo. The quote was transcribed from the article Negative by D’Alembert (1717-1738) for Diderot Encyclopedia.
In order to be able to determine the whole notion, we must see, first, that those so called negative quantities, and mistakenly assumed as below Zero, are quite often represented by true quantities, as in Geometry where the negative lines are no different from the positive ones, if not by their position relative to some other line or common point. See CURVE. Therefore, we may readily infer that the negative quantities found in calculation are, indeed, true quantities, but they are true in a different sense than previously assumed. For instance, assume we are trying to determine the value of a number x which, added to 100, gives 50, Algebra tells us that: x + 100 = 50, and that: x = –50, showing that the quantity x is equal to 50, and that instead of being added to 100, it must be subtracted from that number. Consequently, the problem should have been stated in the following way: Find the quantity x which, subtracted from 100, gives 50. Thus, we would have: 100 – x = 50, and x = 50. The negative form for x would then no longer exist. Thus, the negative quantities really show the calculation of positive quantities assumed in a wrong position. The minus sign found in front of a quantity is meant to rectify and correct a mistake in the hypothesis, as clearly shown by the above example. (quoted in Glaeser, 1981, 323–324)
Interesting, isn’t it? Numbers are abstract ideas. They get their meanings from the context we apply them to. Of course from the school mathematics point of view we cannot start with this idea.
Here are the different meanings of the negative number that students should know before they leave sixth grade: 1) it is the result of subtraction when a bigger number is taken away from a smaller number; 2) it is the opposite of a counting/ natural number; 3) that when added to its opposite counting number results to zero; and 4) it represents the position of a point to the left of zero.
Likewise for the minus sign which indicates subtraction. Subtraction has three meanings: take away, find the difference, and inverse operation of addition. For further explanation read the post What exactly are we doing when we subtract?
