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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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 Jonathan J. Crabtree Posts: 355 From: Melbourne Australia Registered: 12/19/10
Re: Proving a definition of multiplication (wrong) by induction
Posted: Jan 29, 2013 5:49 PM

> 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

Dear GSC

Thank you for your interest and taking the time to document your thoughts.

Rather than repeat the logic, please refer to my extended response to Dave.

Jonathan
P.S. Best wishes from Australia, where like India, we also celebrated 26th January as our national day.