Chapter V: Negative Numbers

61. _Negative Numbers_ may be regarded as resulting from the commutative law for addition and subtraction. According to this law, 10 + 3 + 6 - 7 = 10 + 3 - 7 + 6 = 3 + 6 - 7 + 10 = &c. But, if we write the expression as 3 - 7 + 6 + 10, this means that we must first subtract 7 from 3. This cannot be done; but the result of the subtraction, if it could be done, is something which, when 6 is added to it, becomes 3 - 7 + 6 = 3 + 6 - 7 = 2. The result of 3 - 7 is the same as that of 0 - 4; and we may write it "-4," and call it a _negative number_, if by this we mean something possessing the property that -4 + 4 = 0.

This, of course, is unintelligible on the grouping system of treating number; on the counting system it merely means that we count backwards from 0, just as we might count inches backwards from a point marked 0 on a scale. It should be remembered that the counting is performed with something as unit. If this unit is A, then what we are really considering is -4A; and this means, not that A is multiplied by -4, but that A is multiplied by 4, and the product is taken negatively. It would therefore be better, in some ways, to retain the unit throughout, and to describe -4A as a _negative quantity_, in order to avoid confusion with the "negative numbers" with which operations are performed in formal algebra.

The positive quantity or number obtained from a negative quantity or number by omitting the "-" is called its _numerical value_.