# Who first defined quantum integers?

Who first gave the definition of quantum integers $$[m]_q = \frac{1 - q^m}{1 - q}$$ and addition as $$[m]_q \oplus_q [n]_q = [m]_q + q^m [n]_q$$ and multiplication as $$[m]_q \otimes_q [n]_q = [m]_q [n]_{q^m}$$ and in which context?

• In the first displayed equation, presumably $m$ is supposed to be $n$. – Joe Silverman Oct 20 '14 at 11:04
• By some reason physicists prefer $\frac{q^m-q^{-m}}{q-q^{-1}}$... – მამუკა ჯიბლაძე Oct 20 '14 at 12:00
• Gauss used Gaussian binomial coefficients, and quantum integers are special cases of these. – Chris Godsil Oct 20 '14 at 12:12
• I believe Gauss introduced Gaussian binomial coefficients in 1811 in Summatio quarumdam serierum singularium. I just flipped quickly through it, and I couldn't see any explicit discussion of quantum integers per se, but Chris Godsil is certainly right that they are implicit since they are the special case $m$ choose $1$ (and in any case I might have missed some broader discussion). – Henry Cohn Oct 20 '14 at 12:22
• See also "The history of q-calculus and a new method" by Ernst citeseerx.ist.psu.edu/viewdoc/… – Tom Copeland Sep 6 '16 at 1:28