- #1
Atran
- 93
- 1
A proper number is expressed in [itex]\pi[/itex] in a similar way as a decimal integer is expressed in base 2. For example, [itex]4375_{\pi} = 4{\pi^3}+3{\pi^2}+7{\pi}+5[/itex]. The only exception I make is that the 10 digits are included when expressing a number with [itex]\pi[/itex]. To clarify, the first positive such numbers are: [itex]0,1,2,3,4,5,6,7,8,9,{\pi},{\pi}+1,{\pi}+2,...,2{\pi},2{\pi}+1,...5{\pi},5{\pi}+1,5{\pi}+2,...,9{\pi}+8,9{\pi}+9,{\pi}^2,...[/itex].
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?
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?