[Math] Help with Proof of the Associative Property of Addition of Complex Numbers

Question

I am trying to derive a proof of the associative property of addition of complex numbers using only the properties of real numbers.

I found the following answer but was hoping someone can explain why it is correct, since I am not satisfied with it (From Using the properties of real numbers, verify that complex numbers are associative and there exists an additive inverse):

\begin{equation}
\begin{aligned}
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{aligned}
\end{equation}

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.

I don't see how step 2 and 3 are commutativity in R. Wouldn't these steps require commutativity and associativity in C as well? For example, in step 2, I am presuming the author of the proof did the following rearrangement:

\begin{equation}
[(c + di) + (e + fi)] \\
[c + di + e + fi] \\
[c + e + di + fi] \\
[(c + e) + (di + fi)] \\
\end{equation}

But doesn't the following rearrangement

\begin{equation}
[(c + di) + (e + fi)] \\
[c + di + e + fi] \\
\end{equation}

require associativity in C? And doesn't the following

\begin{equation}
[c + di + e + fi] \\
[c + e + di + fi] \\
\end{equation}

require commutativity in C?

There are some other steps I am confused by as well, like step 4 only requiring associativity in R when clearly there is an associativity in C occurring on the right end of the equation.

Can anyone provide insight into why this proof is correct?

Answer 1

Below I've annotated the proof with justifications for each equality.

$$\begin{aligned} &\quad\ \ z_1 + (z_2 + z_3)\\[.2em] &= (a + bi) + [(c+di) + (e +fi)]\quad \:\!\text{by definition of a complex number} \\[.2em] &= (a+bi)+[(c+e)+(d+f)i]\quad\,\ \text{by definition of addition in}\ \Bbb C\\[.2em] &= [a+(c+e)]+[(b+(d+f))i]\, \quad \text{by definition of addition in}\ \Bbb C\\[.2em] &= [(a+c)+e]+[((b+d)+f)i] \,\quad \text{by associativity of addition in}\ \Bbb R\\[.2em] &= [(a+c)+(b+d)i)]+(e+fi) \quad \text{by definition of addition in}\ \Bbb C\\[.2em] &= [(a+bi)+(c+di)]+(e+fi) \quad \text{by definition of addition in}\ \Bbb C\\[.2em] &= (z_1+z_2)+z_3 \end{aligned}$$