``God made the integers; all else is the work of man.'' --Kronecker
Denote the equivalence class containing (a,b) by [(a,b)] Define addition by
Define multiplication by
Denote the equivalence class containing a by [a]. Define addition by
Define multiplication by
Denote the equivalence class containing (a,b) by [(a,b)]. Define addition by
Define multiplication by
Let X be the set of Cauchy sequences of rational numbers. Define equivalence by
Denote the equivalence class containing by . Define addition by
Define multiplication by
Denote the equivalence class containing (a,b) by [(a,b)]. Define addition by
Define multiplication by
In each of the above cases, you should be able to verify that
Are you convinced that each of the above structures is isomorphic to the number systems as we usually envision them?