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So it's easy to show that the rationals and the integers have the same size, using everyone's favorite spiral-around-the-grid.

Can the approach be extended to say that the set of complex numbers has the same cardinality as the reals?

Asaf Karagila
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Ethan
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4 Answers4

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Yes.

$$|\mathbb R|=2^{\aleph_0}; |\mathbb C|=|\mathbb{R\times R}|=|\mathbb R|^2.$$

We have if so:

$$|\mathbb C|=|\mathbb R|^2 =(2^{\aleph_0})^2 = 2^{\aleph_0\cdot 2}=2^{\aleph_0}=|\mathbb R|$$

If one wishes to write down an explicit function, one can use a function of $\mathbb{N\times 2\to N}$, and combine it with a bijection between $2^\mathbb N$ and $\mathbb R$.

Asaf Karagila
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15

Of course. I will show it on numbers in $[0,1)$ and $[0,1)\times[0,1)$. Consider $z=x+iy$ with $x=0.x_1x_2x_3\ldots$ and $y=0.y_1y_2y_3\ldots$ their decimal expansions (the standard, greedy ones with no $9^\omega$ as a suffix). Then the number $f(z)=0.x_1y_1x_2y_2x_3y_3\ldots$ is real and this map is clearly injective on the above mentioned sets. Generalization to the whole $\mathbb C$ is straightforward. This gives $\#\mathbb C\leq\#\mathbb R$. the other way around is obvious.

J. W. Tanner
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yo'
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    This requires a bit more work. The map isn’t well-defined until you deal with the $0.4999\dots=0.5000\dots$ issue; if you deal with that straightforwardly, it’a nor surjective. – Brian M. Scott Nov 26 '12 at 19:14
  • Yes, you are right. However, they all all (complex) rational hence of no interest for the sets of continuum cardinality. I'll add a comment. – yo' Nov 26 '12 at 19:16
  • And btw, usually a string with suffix $9^\omega$ is not considered to be an _expansion_ (it is only a _representation_), in the usual greedy expansions as defined by Rényi in 1957. – yo' Nov 26 '12 at 19:22
  • I’ve never seen anyone make a distinction between *representation* and *expansion*, and I very much doubt that the distinction can be considered standard; certainly it does not qualify as well-known, so if you use it, you need to explain it. – Brian M. Scott Nov 26 '12 at 19:29
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    And yes, I know that only countably many numbers are affected and that this does not affect the result, but I don’t know that the OP knows this. – Brian M. Scott Nov 26 '12 at 19:33
  • The sequence $a_1a_2a_3\ldots$ _represents_ $x$ if $x=\sum a_i\beta^{-i}$ (let's say it is a property of the string), you can speak about a greedy one (maximal in the lexicographical order) etc., but then, you only add more properties. The _Rényi expansion_ of $x$ is the one given by $x_0=x$, $a_i=\lfloor \beta x_{i-1}\rfloor$, $x_i=\beta x_{i-1}-a_i$, which is an _algorithm_ for obtaining the representations, and then you can speak about _expanding_ the number. The fact that such representation is greedy (in the sense of maximality w.r.t. some order) has to be proved. – yo' Nov 26 '12 at 19:35
  • Yes, I inferred all that from your earlier comments. It doesn’t affect mine. – Brian M. Scott Nov 26 '12 at 19:37
  • your statement "the other way around is obvious" might be lost on some beginners. can you show how $f$ is surjective? – chharvey Sep 27 '17 at 02:01
  • This argument seems strange to me. Let me show why. Consider a real number $a$ with $\ldots x_2 x_1 x_0 . y_0 y_1 y_2 \ldots$ its decimal expansion. Then the number $f(a) = \ldots x_2 y_2 x_1 y_1 x_0 y_0$ is an integer, right? Thus $\# \mathbb R \le \# \mathbb Z$. Do you see the issue? – DaBler Oct 15 '18 at 14:55
  • @DaBler That doesn't work in the case you mention; integers have to have finite length as strings, so your f(a) isn't actually an integer, or even a number of any sort. But decimal expansions have countably infinite length, so this argument (modulo the issues mentioned earlier around non-unique representation) works fine for interweaving two decimal expansions. Or any number of expansions, for that matter. – Justin Hilyard Aug 12 '19 at 18:21
4

Consult #4b in http://faculty.lasierra.edu/~jvanderw/classes/m415a03/hw8ans.pdf.

A straightforward bijection $B : \mathbb{R}^2 \rightarrow \mathbb{C}$ is: $B(a,b) = a + bi$. I omit the verification of injectivity and surjectivity. Then $|C| = |\mathbb{R}^2|$. The separate result that $|\mathbb{R}^k| = |\mathbb{R}| \; \forall \; k \in \mathbb{N}$ implies $|\mathbb{R}^2| = |\mathbb{R}|$. Altogether, $|\mathbb{C}| = |\mathbb{R}^2| = |\mathbb{R}|$.

joseville
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One particularly nice class of bijections from $\Bbb R$ to $\Bbb C = \Bbb R^2$, which is in my opinion a little bit similar to the spiral around the grid, is given by the space-filling curves.

Dominik
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    This is incorrect. Space filling curves are not injective. – Dan Rust Aug 08 '14 at 15:01
  • @DanRust: Can you explain why? – DaBler Oct 15 '18 at 14:58
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    If an injective and surjective curve in the square existed, then this would imply that such a curve is in fact a homeomorphism, as the interval is compact, and the square is Hausdorff. We know they are not homeomorphic as the interval has a cut point and the square does not. – Dan Rust Oct 15 '18 at 15:05