#### Answer

The statement is true. See below for proof.

#### Work Step by Step

We are given that $z=a+bi$:
$z\cdot\overline{z}=(a+bi)\cdot\overline{(a+bi)}=(a+bi)(a-bi)=a^{2}-abi+abi-b^{2}i^{2}=a^{2}+b^{2}$
We know that $a$ and $b$ are real numbers, so $a^2+b^2$ is a real number.