# Complex Sums

# Why

We want to extend addition to $\C $.

Let $z_1, z_2 \in \C $ with $z_1 = (x_1,
y_1)$ and $z_2 = (x_2, y_2)$.
The complex sum of $z_1$
and $z_2$ is the complex number
$(x_1+x_2,y_1+y_2)$.

## Notation

For $z_1, z_2 \in \C $, we denote the complex
sum of $z_1$ and $z_2$ by $z_1 + z_2$.
The notation is justified because the complex
sum of two purely real complex numbers
corresponds to the purely real complex numbers
whose real part is the real sum of the real
parts of the first two numbers.

Recall that we denote $z_1 = x_1 + iy_1$ and
$z_2 = x_2 + iy_2$.
For example, we can express the definition of
addition as

\[
z_1 + z_2 = (x_1 + x_2) + i(y_1 + y_2).
\]

# Properties

For all $z_1, z_2 \in \C $, we have $z_1 +
z_2 = z_2 + z_1$.

For all $z_1, z_2, z_3 \in \C $, we have and
$z_1 + (z_2 + z_3) = (z_1 + z_2) + z_3$.

# Complex addition

We call the operation that associates a pair
of complex numbers with their sum
complex addition.
The operation is symmetric (commutative).

# Additive identity and inverse

Notice that the complex number $(0, 0)$ is the
additive identity.
It is unique,
and so we call it the complex
multiplicative identity.
Likewise, notice that additive inverse for $(x,
y)$ is $(-x, -y)$.