I am self-learning, so I need guidance, as I am unsure whether my approach is sufficient. There are two questions, both asking to verify a property of the complex numbers using the properties of real numbers.
(1) Verify that $z_{1}+(z_{2}+z_{3})=(z_{1}+z_{2})+z_{3}$ for all $z_{1}, z_{2}, z_{3} \in \mathbb{C}$
My approach:
\begin{align}z_{1}+(z_{2}+z_{3}) &= (a+bi)+[(c+di)+(e+fi)]\\
&=(a+bi)+[(c+e)+(di+fi)]\\
&=[a+(c+e)]+[bi+(di+fi)]\\
&=[(a+c)+e]+[(bi+di)+fi]\\
&=[(a+c)+(bi+di)]+(e+fi)\\
&=[(a+bi)+(c+di)]+(e+fi)\\
&=(z_{1}+z_{2})+z_{3}
\end{align}
I justify step 1 by the definition of complex numbers, step 2 and 3 by commutativity in R, step 4 by associativity in R, step 5 and 6 by commutativity in R.
(2) Additive inverse for every $z\in\mathbb{C}$, $\exists$w ∈ C: z + w = 0.
My approach:
Let $w=|(-z)|$. By the definition of complex numbers $\left|(-z)\right|=\left|-(a+bi)\right|$, and we have \begin{align}\left|(-z)\right| &= \left|-(a+bi)\right|\\ &=\left|-a-bi\right|\\ &=\sqrt{(-a)^2-(bi)^2}\\ &=\sqrt{a^2-b^2*i^2}\\ &=\sqrt{a^2-b^2*-1}\\ &=\sqrt{a^2+b^2}\\ &=\left|a+bi\right|\\ &=\left|z\right|\\ \end{align}
so there exists a number $w\in\mathbb{C}$ such that $z+w=0$.