Complex Addition is Associative

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Theorem

The operation of addition on the set of complex numbers $\C$ is associative:

$\forall z_1, z_2, z_3 \in \C: z_1 + \left({z_2 + z_3}\right) = \left({z_1 + z_2}\right) + z_3$


Proof

From the definition of complex numbers, we define the following:

$z_1 = \left({x_1, y_1}\right)$
$z_2 = \left({x_2, y_2}\right)$
$z_3 = \left({x_3, y_3}\right)$

where $x_1, x_2, x_3, y_1, y_2, y_3 \in \R$.


Thus:

\(\displaystyle z_1 + \left({z_2 + z_3}\right)\) \(=\) \(\displaystyle \left({x_1, y_1}\right) + \left({\left({x_2, y_2}\right) + \left({x_3, y_3}\right)}\right)\) $\quad$ Definition 2 of Complex Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({x_1, y_1}\right) + \left({\left({x_2 + x_3, y_2 + y_3}\right)}\right)\) $\quad$ Definition of Complex Addition $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({x_1 + \left({x_2 + x_3}\right), y_1 + \left({y_2 + y_3}\right)}\right)\) $\quad$ Definition of Complex Addition $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({\left({x_1 + x_2}\right) + x_3, \left({y_1 + y_2}\right) + y_3}\right)\) $\quad$ Real Addition is Associative $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({\left({x_1 + x_2, y_1 + y_2}\right)}\right) + \left({x_3, y_3}\right)\) $\quad$ Definition of Complex Addition $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({\left({x_1, y_1}\right) + \left({x_2, y_2}\right)}\right) + \left({x_3, y_3}\right)\) $\quad$ Definition of Complex Addition $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({z_1 + z_2}\right) + z_3\) $\quad$ Definition 2 of Complex Number $\quad$

$\blacksquare$


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