Date: Jan 28, 2013 11:29 PM
Author: GS Chandy
Subject: Re: Proving a definition of multiplication (wrong) by induction
Jonathan Crabtree posted Jan 29, 2013 6:47 AM (GSC's remark interspersed):
> 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.
To the best of my understanding, the definition does NOT fail proof by induction (see attachment, using "n" instead of "b", for convenience, and showing start of proof only for integers). [My 'formal statements' in the document won't pass muster with teachers demanding a high degree of rigor and precision, but I'm unable to do anything about that at this point of time].
In my opinion, the Collins dictionary definition fails mainly because it is rather poorly articulated.
[I've not done anything with your P.S.].
> So what other proofs can be used to prove ab does not
> equal a added to itself b times?
> Jonathan Crabtree
> 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
Message was edited by: GS Chandy