[OT] Number theory [Was: A use for integer quotients]

[Michael Abbott]

"michael" <serrano at ozemail.com.au> wrote in
news:IjA77.97824$Rr4.589251 at ozemail.com.au: 

> 
> So strictly speaking, Z (the set of integers) is not a subset of R (the
> set of reals)?
> 

That's absolutely right.  There's an injective function from Z to R which 
preserves the ring and order structure on Z; similarly, the rationals Q can 
be injectively embedded in the reals:

    	Z >-> Q >-> R

In this case, the map Q to R also preserves the field structure.

In both cases, of course, there is structure which is not preserved.  For 
example, integer division (eg, 1/2 = 0 remainder 1) doesn't really make all 
that much sense in R except by appealing to its original formation in Z.  
And conversely, of course, Q and R add extra structure.

Actually of course, there's a whole family of fields between Q and R.

Finally, just to remind people that embedding can lose structure as well as 
add it, consider the well known embedding

    	R >-> C

of the reals into the complex numbers.  The order relation x <= y cannot be 
sensibly defined on C, so order structure is lost by this embedding.