James Harris wrote: > > ... > > Since you guys believe in fractions, I've also said at times that > they're in a "flat" ring, which is just to say that you don't have any > fractions in it.
What's a ring without fractions in it? Is it a ring with unity in which, for every x != 1, there is no y such that x*y = 1 or y*x = 1; and for x = 1 the only such y is 1?
Whatever it is, are you defining "flat ring" to mean "ring without fractions"?
> The confusion on this issue can be demonstrated by (1+sqrt(3)i)/2, > which at least one poster BELIEVED was a fraction (hey, it LOOKS like > a fraction) though it's provably not.
So define "fraction" and prove it. Or define "fraction" and leave someone else to prove (or otherwise) it.
> Now, some have claimed that you can just use algebraic numbers as a > grab bag to handle this and all cases like it, but I think my example > above shows that doesn't work. Besides, I think the definition for > algebraic numbers is circular and hardly useful anyway.
Why do you think that the definition of algebraic numbers is circular? "Hardly useful" I have no objection to; if you hardly use them, so be it. But circular is another matter.
> > ... > > My issue with the current view in the field is that you can talk about > rings where "factor" is meaningless. If you can do that then what's > the point of the ring?
Do you mean that in the reals (for instance) everything is a factor of everything? (Leaving aside zero.) That doesn't make "factor" meaningless, though it might make it useless.
> > Since I only need the operations of addition and multiplication, and > start from a flat ring--integers--there's NO WAY I can end up in a
Is that a definition of "flat ring"? I mean, is "flat ring" just another name for the ring of integers?
> ring where factors don't matter. > > ... > > So, let me re-emphasize my point, by asking you a simple question: > > Can you start with integers and end up in the field of rationals with > fractions...your regular old garden variety fraction like 1/2...using > only addition and multiplication?
The integers are closed under addition and multiplication, so no, you can't get 1/2 by iterating the ring operations. But then all rings are closed under the ring operations.
When you write "fraction", do you mean "non-integral rational number"?
> > If you answer no, then logic will drive you to accept my proof. > > If you answer yes, then I want you to demonstrate. > > If you refuse to answer "yes" or "no", then you'll say volumes to the > world which is suddenly very attentive to what's happening here. > > Yes--the world--so be honest. > > James Harris