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Topic: Proving a definition of multiplication (wrong) by induction
Replies: 19   Last Post: Feb 8, 2013 2:36 AM

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 GS Chandy Posts: 8,307 From: Hyderabad, Mumbai/Bangalore, India Registered: 9/29/05
Re: Proving a definition of multiplication (wrong) by induction
Posted: Jan 28, 2013 11:29 PM
 Math-teach - Proof of multiplication.doc (21.5 K)

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