Is C bigger than R?

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?

marcus

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.......

MathematicalPhysicist

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

Big-T

How could that possibly be a bijection? Obviously, [tex]z_1=a+ib[/tex] is mapped to the same point as [tex]z_2=a z_1[/tex], 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 [tex]\mathbb{R}^n[/tex].

MathematicalPhysicist

Correct Big-T, but at least it's onto.
(-:

adriank

|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.

Big-T

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.