Complex Addition is Commutative

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Theorem

The operation of addition on the set of complex numbers is commutative:

$\forall z, w \in \C: z + w = w + z$


Proof

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

$z = \left({x_1, y_1}\right)$
$w = \left({x_2, y_2}\right)$

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


Then:

\(\displaystyle z + w\) \(=\) \(\displaystyle \left({x_1, y_1}\right) + \left({x_2, y_2}\right)\) $\quad$ Definition 2 of Complex Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({x_1 + x_2, y_1 + y_2}\right)\) $\quad$ Definition of Complex Addition $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({x_2 + x_1, y_2 + y_1}\right)\) $\quad$ Real Addition is Commutative $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({x_2, y_2}\right) + \left({x_1, y_1}\right)\) $\quad$ Definition of Complex Addition $\quad$
\(\displaystyle \) \(=\) \(\displaystyle w + z\) $\quad$ Definition 2 of Complex Number $\quad$

$\blacksquare$


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