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