Virgil
Posts:
4,483
Registered:
1/6/11
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Re: Bijection Between Complex Numbers and Real Numbers
Posted:
Feb 19, 2012 3:16 PM
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In article
<cb39da8d-c390-4a10-a159-2fdaa5c7e22a@l14g2000vbe.googlegroups.com>,
Pubkeybreaker <pubkeybreaker@aol.com> wrote:
> On Feb 19, 12:03 am, Michael Ejercito <mejer...@hotmail.com> wrote:
> > What bijective function exists such that every complex number maps
> > to a unique real number, and likewise every real number maps to a
> > unique complex number?
> >
> > Michael
>
> Consider a bijective map from [0,1] to the unit square.
>
> Let r \in [0,1]. Let (x,y) be a point in the unit square.
> let r = .a1 a2 a3 .....
>
> Let x = .a1 a3 a5 .....
> y = .a2 a4 a6 .....
>
> i.e. take x as being formed from every other digit in the decimal
> representation of r. Similarly for y.
That maps the single point r = 0.1000... = 0.0999...
to both the point x = 0.1000..., y = 0.000...
and to the point x = 0.0999... = 0.01 , y = 0.999... = 1, and maps those
two complex points back to a single real point.
And similarly for all the infinitely many reals in [0,1] having dual
representation.
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