Rational Arithmetic
Why
What are addition and multiplication for rationals? What are the identity elements?
Definition
We call the operation of forming rationals sums rational addition. We call the operation of forming rational products rational multiplication.
Results
It is easy to see the following.1
The additive identity for $\Q $ is
$\eqc{(0_{\Z }, 1_{\Z })}$.
The multiplicative identity for $\Z $ is
$\eqc{(1_{\Z }, 1_{\Z })}$.
Notation
We denote the additive identity of $\Q $ by $0_{\Q }$ and the multiplicative identity by $1_{\Q }$. We denote the set $\Set{q \in \Q }{q \geq 0_{Q}}$ by $\Q _{+}$.
Distributive
For rationals $x, y, z \in \Z $, $x \cdot (y
+ z) = x\cdot y + x\cdot z$.2