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In message <> you wrote:

> On Thursday 19 May 2005 5:52 am, Gavin Wraith wrote:
> > The remainder theorem can be articulated in different
> > ways and the definition of "remainder" depends on which
> > you use. For example:
> am i the only one that thinks it's weird to discuss so long about 
> remainder/modulus when there's no integer division?

But the ring of integers does not have division! That is the whole
point, and is the reason why you have to specify what you mean by
the word. In a system with multiplication (*) to say that x divides y
means that for some z, y can be expressed as x*z. If z happens to be
unique, then it makes sense to write z as y/x. For integers you have
uniqueness if x is nonzero, but you do not necessarily have existence.
For real numbers you do have existence, when y is nonzero. The notion
of "integer division" is Procrustean; it is forcing a square peg into
a round hole. Google the giant Procrustes if you do not know about
his unsavoury habits.
Gavin Wraith (
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