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Topic: Bijection Between Complex Numbers and Real Numbers
Replies: 59   Last Post: Feb 24, 2012 7:12 AM

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 Tommy Jensen Posts: 132 From: Daegu, Korea Registered: 12/6/09
Re: Bijection Between Complex Numbers and Real Numbers
Posted: Feb 23, 2012 9:37 PM

> > Do you have an equally pretty bijection from
> > from (0,1] to |R, which
> > completes the construction?
> > =kluto

>
> Let f(x) = (1-2x)/(x*(1-x)), and note that f maps
> the
> open interval (0,1) bijectively to the set of reals.

Or use cot, which seems tailored for this.

> Let g(x) = x provided x is NOT of the form 1/n for n
> a positive integer, and let g(1/n) = 1/(n+1) for any
> positive integer n.
>
> Note that g : (0,1] -> (0,1) is a bijection.
>
> So the composition f o g gives a bijection between
> (0,1] and the set of reals.
>
> Not very "pretty" I admit.

It is a cute application of Hilbert's hotel though.
It is nice, thanx!

=kluto

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