Virgil
Posts:
8,833
Registered:
1/6/11


Re: Bijection Between Complex Numbers and Real Numbers
Posted:
Feb 19, 2012 3:16 PM


In article <cb39da8dc3904a10a1592fdaa5c7e22a@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. 

