So I have this algebraic structure defined on complex numbers: $z_1\star z_2=z_1+z_2-z_1 \cdot z_2$
At first exercise I found the identity element like this: $e=e_1+e_2i$ such that $e\star z=z\star e=z$ and I found $e_1=e_2=0$ so the identity element is $0$.
At the second exercise, I need to find the symmetric of $i$ (the right answer is $\frac{1-i}{2}$ )
I started like in the first exercise but I got stuck. I took $z'=a'+b'i$ such that $z'\star z=z\star z'=0$ and I made a system but I don't get anything useful. How to start?
I am assuming you are looking for the inverse of $i$ under the given operation. Let $w$ be the inverse of $i$, then
\begin{align*} w \star i & =0\\ w+i -iw & =0\\ w & = \frac{i}{i-1}\\ & =\frac{i(-i-1)}{(i-1)(-i-1)}\\ & = \frac{1-i}{2} \end{align*}
For the identity element. Say $e$ is the identity element, then we want $z \star e=e \star z=z$ for all $z$. So we can choose any $z$ we want and same $e$ should work. For example, we can choose $z=0$, to get $0+e-(0)(e)=0$. This suggests that $e=0$ can possibly be an identity element. Now we can verify that $e=0$ works for all $z$ because
$$0 \star z =0+z-(0)(z)=z.$$