# Complex Addition is Commutative

## 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:

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

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

Then:

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

$\blacksquare$

## 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:

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

$\blacksquare$