Explore the fascinating world of alternating and symmetric polynumbers in this 28-minute mathematics lecture. Delve into the introduction of subtraction in arithmetic with integral polynumbers, building upon the concept of negative numbers through the duality of msets and anti-msets. Discover the fundamental reflection symmetry denoted by sigma, which allows for the definition of symmetric and anti-symmetric (alternating) polynumbers. Investigate the basis B_n of symmetric polynumbers and the closely related basis A_n of alternating polynumbers, uncovering their algebraic relationships. Learn about the significant role of central polynumbers C_n in various mathematical areas, including q-series, representations of SU(2), hypergroups, quantum groups, and physics. Gain insights into this often-overlooked aspect of elementary algebra, which offers a fresh perspective beyond the traditional focus on functions in mathematical education.