Real Sums
Why
We want to add real numbers.1
Definition
The real sum of two
real numbers $R$ and $S$ is the set
\[
\Set{t \in \Q }{\exists r \in R, s \in S \text{ with } t =
r + s}.
\]
Notation
Suppose $x, y \in \R $ are two numbers. We denote the sum of $x$ and $y$ by $x + y$.
Properties
We can show the following.2
$x + (y + z) = (x + y) + z$
$x + y = y + x$
The set of negative rational numbers is the
additive identity.
We denote the additive identity of $\R $ under $+$ by $0_{\R }$. When it is clear from context, we call $0_{\R }$ “zero”.