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Re: Bijection Between Complex Numbers and Real Numbers
Posted:
Feb 20, 2012 7:20 AM
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In article <513f89be-feb4-4e36-8210-8f2130314b70@x19g2000yqh.googlegroups.com>,
Michael Ejercito <mejercit@hotmail.com> wrote:
>> Though this has the usual problem with multiple representations. For
>> example, both 0.1 and 0.00909090... map to (0.1, 0).
> That is correct. This would imply that each complex number maps
>onto MULTIPLE real numbers, and certainly the reals can not have a
>GREATER cardinality than the complexes.
Given a surjection both ways, the dual Schroder Bernstein theorem
shows that there is a bijection. This however relies on the axiom
of choice.
If we take the interleaved digits as a base-11 number we have an injection
both ways, and standard Schroder Bernstein gives us a bijection.
-- Richard
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