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It is well known that there is a one-to-one mapping between the integers and the rationals.

It is also well known that there no such mapping between the rationals and the real numbers.

Is there a one-to-one correspondence between the set of real numbers and the set of complex numbers?

As part of the answer, could you either provide a high-level sketch or a reference (either the name of a famous theorem or a url) that would help me to understand the argument.

Larry Freeman
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    I think I heard that there is a bijection between $\mathbb{R}$ and $\mathbb{R}^2$ - this would imply the bijection between the reals and the complex numbers, but I think I also recall hearing that constructing such bijection is not very pretty. See http://mathoverflow.net/questions/126069/bijection-from-mathbbr-to-mathbbr2 –  Jan 13 '16 at 16:03

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Yes there is. Actually, the is a bijection between $\Bbb R$ and $\Bbb R^n$ for any $n\in \Bbb N$. You can think of $\Bbb C$ as $\Bbb R^2$ in this case. It is called Cardinality of the continuum, proven by Cantor.

The proof dates back to Cantor's correspondence with Dedekind in the 1870s. The result was really shocking at the time since it was tacitly assume that there can be no bijection between manifold with different dimension. Dedekind resolve the problem by conjecturing that no such continuous bijection exists. The fact was later proved by Brouwer (if I recall correctly) many decades after that.

BigbearZzz
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For any infinite set $X$ and any finite set $I$, $X^I$ is equipotent to $X$

Bernard
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