59. Results of Inverse Operations.—Addition, multiplication and involution are direct processes; and, if we start with positive integers, we continue with positive integers throughout. But, in attempting the inverse processes of subtraction, division, and either evolution or determination of index, the data may be such that a process cannot be performed. We can, however, denote the result of the process by a symbol, and deal with this symbol according to the laws of arithmetic. In this way we arrive at (i) negative numbers, (ii) fractional numbers, (iii) surds, (iv) logarithms (in the ordinary sense of the word).
60. Simple Formulae.—The following are some simple formulae which follow from the laws stated in § 58.
(i) (a + b + c + ...)(p + q + r + ...) = (ap + aq + ar + ...) + (bp + bq + br + ...) + (cp + cq + cr + ...) + ...; i.e. the product of two or more numbers, each of which consists of two or more parts, is the sum of the products of each part of the one with each part of the other.
(ii) (a + b)(a − b) = a2 − b2; i.e. the product of the sum and the difference of two numbers is equal to the difference of their squares.
(in) (a + b)2 = a2 + 2ab + b2 = a2 + (2a + b)b.
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.
VI. Fractional and Decimal Numbers