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I'm having trouble coming up with an easy-to-describe surjection $f : \mathbb{R} \to \mathbb{C}$. Here's what I came up with: (Edit: this doesn't work; see Qiaochu's comment)

Define $P(x \in \mathbb{R}) = \frac{1}{1-e^{-x}}$ and $P^{-1}(x \in (0,1)) = -\log \frac{1-x}{x}$.

Let $f(x) = P(\operatorname{even}(P^{-1}(x))) \cdot e^{2 \pi i \operatorname{odd}(P^{-1}(x))}$, where

$$\operatorname{even}(0.b_0b_1b_2\ldots) = 0.b_0b_2b_4\ldots, \\\operatorname{odd}(0.b_0b_1b_2\ldots) = 0.b_1b_3b_5\ldots.$$

However, this seems needlessly complicated. Is there a simpler way?

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    $\text{even}$ and $\text{odd}$ are not well-defined functions (consider numbers that have more than one decimal expansion). You are much better off finding a bijection from $\mathbb{R}$ to a more manageable set first (I suggest $2^{\mathbb{N}}$).2012-12-02
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    Even and odd are well-defined if you always use the finite expansion of numbers in $10^m\mathbb{Z}$ for $m\in \mathbb{Z}$. (The usual "greedy" expansion.) That the results of even and odd are not always of this form is no problem. In fact when you extend them to include digits in front of the decimal point you can use them as real and imaginary parts of your function.2012-12-02
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    @WimC: I'd consider your last sentence an easy-to-describe surjection. Can you post this as an answer?2012-12-02
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    @Snowball That wouldn't add much to the existing answer and I did not describe what happens to negative numbers. It's not complete yet.2012-12-02

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