Is $(a+bi)(a-bi) = a^2 + b^2 $ solely a real number or a complex number? I have not dealt with complex numbers for a while now, but I was wondering if I multiplied the complex number $a+bi$ by its conjugate $a-bi$ to obtain
$$(a+bi)(a-bi) = a^2 + b^2 $$
where $a,b \in \mathbb{R}$, does $a^2 + b^2 \in \mathbb{R}$ or $\mathbb{C}$?
I am pretty sure that the above product is solely a real number since the imaginary number $i$ has been removed, but I want to make sure.

Yes, it is always in $\mathbb R$. In fact it is the square of the length of the vector pointing to the point in the complex plane.
