(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 × 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 × c × d × e ÷ f = a ÷ (b ÷ c) × (d × e ÷ f).
(v) Commutative Law for Multiplications and Divisions, that multiplications and divisions may be performed in any order: e.g. a ÷ b × c × d = a × c ÷ b × d = a × d × 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.