Complex Addition is Associative

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Theorem

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

$\forall z_1, z_2, z_3 \in \C: z_1 + \paren {z_2 + z_3} = \paren {z_1 + z_2} + z_3$


Proof

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

\(\displaystyle z_1\) \(:=\) \(\displaystyle \tuple {x_1, y_1}\)
\(\displaystyle z_2\) \(:=\) \(\displaystyle \tuple {x_2, y_2}\)
\(\displaystyle z_3\) \(:=\) \(\displaystyle \tuple {x_3, y_3}\)

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


Thus:

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

$\blacksquare$


Examples

Example: $\paren {5 + 3 i} + \paren {\paren {-1 + 2 i} + \paren {7 - 5 i} } = \paren {\paren {5 + 3 i} + \paren {-1 + 2 i} } + \paren {7 - 5 i}$

Example: $\paren {5 + 3 i} + \paren {\paren {-1 + 2 i} + \paren {7 - 5 i} }$

$\paren {5 + 3 i} + \paren {\paren {-1 + 2 i} + \paren {7 - 5 i} } = 11$


Example: $\paren {\paren {5 + 3 i} + \paren {-1 + 2 i} } + \paren {7 - 5 i}$

$\paren {\paren {5 + 3 i} + \paren {-1 + 2 i} } + \paren {7 - 5 i} = 11$


As can be seen:

$\paren {5 + 3 i} + \paren {\paren {-1 + 2 i} + \paren {7 - 5 i} } = \paren {\paren {5 + 3 i} + \paren {-1 + 2 i} } + \paren {7 - 5 i}$

$\blacksquare$


Also see


Sources