## 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:

 $\ds z$ $:=$ $\ds \tuple {x_1, y_1}$ $\ds w$ $:=$ $\ds \tuple {x_2, y_2}$

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

Then:

 $\ds z + w$ $=$ $\ds \tuple {x_1, y_1} + \tuple {x_2, y_2}$ Definition 2 of Complex Number $\ds$ $=$ $\ds \tuple {x_1 + x_2, y_1 + y_2}$ Definition of Complex Addition $\ds$ $=$ $\ds \tuple {x_2 + x_1, y_2 + y_1}$ Real Addition is Commutative $\ds$ $=$ $\ds \tuple {x_2, y_2} + \tuple {x_1, y_1}$ Definition of Complex Addition $\ds$ $=$ $\ds w + z$ 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:

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

$\blacksquare$