Date: Jan 28, 2013 8:17 PM
Author: Jonathan Crabtree
Subject: Proving a definition of multiplication (wrong) by induction
Multiplication* an arithmetical operation, defined initially in terms of repeated addition, usually written a × b, a.b, or ab, by which the product of two quantities is calculated: to multiply a by positive integral b is to add a to itself b times.
i.e. ab = a added to itself b times
This definition fails proof by induction.
So what other proofs can be used to prove ab does not equal a added to itself b times?
P.S. Apart from proof by common sense. Eg.
let a = 1 and b = 0
so 1 x 0 = 1 + 0 (by definition)
and 1 x 0 does not equal 1
let a = 1 and b = 1
so 1 x 1 = 1 + 1 (by definition)
and 1 x 1 does not equal 2