Real and Complex Numbers are Equinumerous

Proof

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

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

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.