Chapter IV: Laws of Arithmetic

58. _Laws of Arithmetic._--The arithmetical processes which we have considered in reference to positive integral numbers are subject to the following laws:--

(i) _Equalities and Inequalities._--The following are sometimes called _Axioms_ (S 29), but their truth should be proved, even if at an early stage it is assumed. The symbols ">" and "<" mean respectively "is greater than" and "is less than." The numbers represented by a, b, c, x and m are all supposed to be positive.

(a) If a = b, and b = c. then a = c;
(b) If a = b, then a + x = b + x, and a - x = b - x;
(c) If a > b, then a + x > b + x, and a - x > b - x;
(d) If a < b, then a + b < b + x, and a - x < b - x;
(e) If a = b, then ma = mb, and a [:] m = b [:] m;
(f) If a > b, then ma > mb, and a [:] m > b [:] m;
(g) If a < b, then ma < mb, and a [:] m < b [:] m.

(ii) _Associative Law for Additions and Subtractions._--This law includes the _rule of signs_, that a - (b - c) = a - b + c; and it states that, subject to this, successive operations of addition or subtraction may be grouped in sets in any way; e.g. a - b + c + d + e - f = a - (b - c) + (d + e - f).

(iii) _Commutative Law for Additions and Subtractions_, that additions and subtractions may be performed in any order; e.g. a - b + c + d = a + c - b + d = a - b + c - b.

(iv) _Associative Law for Multiplications and Divisions._--This law includes a rule, similar to the rule of signs, to the effect that a[:](b[:]c) = a [:] b [X] c; and it states that, subject to this, successive operations of multiplication or division may be grouped in sets in any way; e.g. a b [X] c [X] d [X] e [:] f = a [:] (b [:] c) [X] (d [X] e [:] f).

(v) _Commutative Law for Multiplications and Divisions_, that multiplications and divisions may be performed in any order: e.g. a [:] b [X] c [X] d = a [X] c [:] b [X] d = a [X] d [X] c [:] b.

(vi) _Distributive Law_, that multiplications and divisions may be distributed over additions and subtractions, e.g. that m(a + b - c) = m.a + m.b - m.c, or that (a + b - c) [:] n = (a [:] n) + (b [:] n) + (c [:] n).

In the case of (ii), (iii) and (vi), the letters a, b, c, ... may denote either numbers or numerical quantities, while m and n denote numbers; in the case of (iv) and (v) the letters denote numbers only.

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 S 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) = a^2 - b^2; i.e. the product of the sum and the difference of two numbers is equal to the difference of their squares.

(iii) (a + b)^2 = a^2 + 2ab + b^2 = a^2 + (2a + b)b.