Real and Complex Numbers are Equinumerous

Theorem

The real numbers \(\mathbb{R}\) have the same cardinality as the complex numbers \(\mathbb{C}\).

Proof

Consider a real number and its (potentially infinite) decimal expansion:

\[ \dots 10^{2}a_{2} + 10a_{1} + a_{0} + 10^{-1}a_{-1} + 10^{2}a_{-2} + \dots\]

Construct a complex number by letting the digits of the real part be the even indexed digits of the original real number, and the imaginary part be the odd indexed digits

\[ (\dots + 10a_{2} + a_{0} + 10^{-1}a_{-2} + \dots) + i(\dots + a_{1} + 10^{-1}a_{-1} + \dots)\]

Such a process gives a bijection from \(\mathbb{R}\) to \(\mathbb{C}\), and thus proves that \(|\mathbb{R}| = |\mathbb{C}|\).

Note that care must be taken to deal with the non unique decimal expansions, however in any case it is easy to choose a canonical form.