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.].

GSC

> So what other proofs can be used to prove ab does not

> equal a added to itself b times?

>

> Thanks

> 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

>

> or

>

> let a = 1 and b = 1

> so 1 x 1 = 1 + 1 (by definition)

> and 1 x 1 does not equal 2

>

> *

> http://www.collinsdictionary.com/dictionary/english/mu

> ltiplication

Message was edited by: GS Chandy