## 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:

 $\ds z_1$ $:=$ $\ds \tuple {x_1, y_1}$ $\ds z_2$ $:=$ $\ds \tuple {x_2, y_2}$ $\ds z_3$ $:=$ $\ds \tuple {x_3, y_3}$

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

Thus:

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

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

$\blacksquare$