I'm a bit of a heretic suggesting arithmetic has been built on shaky foundations, yet Euclid can be forgiven as he wrote prior to the acceptance of both zero as a number and and negative numbers. He would have learnt much from Diophantus and more from Brahmagupta of Indian fame.
I have finally found another mathematician who has stated that mk = m added to itself k-1 times.
His name is S. F. Lacroix, whio wrote about multiplcation, "... by adding the number to itself as many times, wanting one, as is to be repeated.
For instance, by the following addition,
16 16 16 16 - -- 64
the number 16 is repeated four times, and added to itself three times." ie k-1 times
So the non Euclidean definition I have created is,
"mk = the combination of m either added to or subtracted from zero k times"
>From this correct definition of multiplication, all rational arithmetic becomes easy, including integer and fraction multiplication. The number lines can now be used for fractions as well as multiplying negatives.
The whole Devlin MIRA debate has been a red herring.
Until now, multiplication itself has never been correctly defined for the rational numbers.
Jonathan Crabtree 1st September 2012
P.S. Think of a photocopying machine. You have one letter and you need four altogether. So what button do you press on the multiplication machine? Three! ie k-1 Press the four button and you end up with five.