- #1

- 93

- 1

Denote this set of numbers of form [itex]a_{n}{\pi}^n+a_{n-1}{\pi}^{n-1}+...+a_{1}{\pi}+a_{0}[/itex] with [itex]Z_{\pi}[/itex]. Obviously [itex]Z_{\pi}[/itex] is ordered, for example [itex]-2{\pi}^2 < 3{\pi}+2[/itex]. [itex]Z_{\pi}[/itex] is countable, since the elements can be arranged in this way: [itex]0,{\pi},-{\pi},2{\pi},-2{\pi},3{\pi},-3{\pi},...[/itex]

Addition and substraction are defined similarly as they are for [itex](Z,+)[/itex]. Following from the definition of a group, [itex](Z_{\pi}, +)[/itex] is clearly a group.

**However, the elements of [itex](Z_{\pi}, +)[/itex] are not rational numbers.**

Does this imply the set of rational integers and irrational integers are isomorphic?