Is C > R? Complex vs Real Set Size

[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]

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...

 

[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]

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 $$\mathbb{R}^n$$.

 

[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.