Additive number theory and the ring of quantum integers

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.