Definition of Multiplication.—We shall define multiplication by the equalities.
(1) a × 1 = a.
(2) a × b = [a × (b − 1)] + a.
Like equality (1), equality (2) contains an infinity of definitions; having defined a × 1, it enables us to define successively: a × 2, a × 3, etc.
Properties of Multiplication.—Distributivity.—I say that
(a + b) × c = (a × c) + (b × c).
We verify analytically that the equality is true for c = 1; then that if the theorem is true for c = γ, it will be true for c = γ + 1.
The proposition is, therefore, demonstrated by recurrence.
Commutativity.—1º I say that
a × 1 = 1 × a.