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:

\(\displaystyle z\) \(:=\) \(\displaystyle \tuple {x_1, y_1}\)
\(\displaystyle w\) \(:=\) \(\displaystyle \tuple {x_2, y_2}\)

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


Then:

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

$\blacksquare$


Examples

Example: $\paren {3 + 2 i} + \paren {-7 - i} = \paren {-7 - i} + \paren {3 + 2 i}$

Example: $\paren {3 + 2 i} + \paren {-7 - i}$

$\paren {3 + 2 i} + \paren {-7 - i} = -4 + i$


Example: $\paren {-7 - i} + \paren {3 + 2 i}$

$\paren {-7 - i} + \paren {3 + 2 i} = -4 + i$


As can be seen:

$\paren {3 + 2 i} + \paren {-7 - i} = \paren {-7 - i} + \paren {3 + 2 i}$

$\blacksquare$


Also see


Sources