I read here: https://proofwiki.org/wiki/Properties_of_Complex_Numbers, that the Complex Numbers are closed under addition and multiplication. I'm having trouble understanding why. I realize this is a fairly simple question and if I missed it in the search please direct me to the correct place.
My thoughts on addition: $\sqrt{-9}-\sqrt{-9}=9i-9i=0$
Is $0$ considered a complex number of the form $a+bi$ where $a,b=0$?
My thoughts on multiplication: $\left(\sqrt{-9}\right)\left(\sqrt{-9}\right)=3i\cdot 3i=9i^2=9(-1)=-9$
And $-9\notin \mathbb C$.
Or, for the same reason as above, is $-9$ considered a complex with $a=-9$ and $b=0$?
Thanks in advanced.
$\mathbb R \subset \mathbb C$, as $a+0i$ is a complex number for all $a \in \mathbb R$. As you suspected, this includes $-9=-9+0i$ and $0=0+0i$, which are complex numbers (and reals, and integers)!