## Is C bigger than R?

Is it true that the set of complex number is bigger than the set of real numbers?

I know that card C = card (R x R) and I think that card (R x R) > card R. Is this true, and if so, why?
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 Quote by samkolb Is it true that the set of complex number is bigger than the set of real numbers? I know that card C = card (R x R) and I think that card (R x R) > card R. Is this true, and if so, why?
I think card (RxR) = card R

I would show this by setting up a one-to-one map between RxR and R

I will just show you a one-to-one between the unit square [0,1]x[0,1] and the unit interval [0,1]
You just look at the two decimal expansions and merge

(0.abcdefg...., 0.mnopqrs....) -> 0.ambncodpeq.......
 C is with cardinality c, or aleph if you want, the same as R. The simple bijection is a+ib |-> (a,b) into RxR. If you want a bijection from C to R, then z=x+iy|->Im(z)/Re(z) it's a bijection to [-infinity,infinity] which is RU{infininity,-infinity} this cardinality is aleph+2=aleph. QED

## Is C bigger than R?

 Quote by loop quantum gravity If you want a bijection from C to R, then z=x+iy|->Im(z)/Re(z) it's a bijection to [-infinity,infinity] which is RU{infininity,-infinity} this cardinality is aleph+2=aleph.
How could that possibly be a bijection? Obviously, $$z_1=a+ib$$ is mapped to the same point as $$z_2=a z_1$$, so it is not an injection.

Marcus has already provided a valid bijection, his "decimal merging" is the classical example of this. Notice how it is also valid in $$\mathbb{R}^n$$.
 Correct Big-T, but at least it's onto. (-:
 |C| = |R2| = |R|. There's some discussion about that in this thread. Minor point: marcus's function isn't even well-defined; consider decimal expansions with infinite trailing "9"s. (For example, 0.0999... = 0.1000..., but (0.0999..., 0.0000...) maps to 0.00909090..., and (0.1000..., 0.0000) maps to 0.10000000... .) However, the mapping from 0.abcdefgh... to (0.acef..., 0.bdfh...) is a well-defined surjection from [0, 1) to [0, 1)2, and that's all you need.
 Marcus' function would be well defined if we agreed to use trailing nines wherever the decimal expansion is terminating, this should of course have been specified.