  Let $m$ and $n$ be positive integers. For the quantum integer $[n]_q = 1 + q
+ ... + q^{n-1}$ there is a natural polynomial addition such that $[m]_q
\oplus_q [n]_q = [m+n]_q$ and a natural polynomial multiplication such that
$[m]_q \otimes_q [n]_q = [mn]_q$. These constructions lead to the construction
of the ring of quantum integers and the field of quantum rational numbers. It
is also shown that addition and multiplication of quantum integers are
equivalent to elementary decompositions of intervals of integers in additive
number theory.
