Explore the fascinating realm of integral polynumbers in this 37-minute mathematics lecture. Delve into arithmetic with multisets, incorporating negative numbers and integral arithmetic using particle/anti-particle duality concepts from physics. Discover two ways to expand polynumber arithmetic: introducing integers as coefficients and as exponents. Examine this larger integral polynomial arithmetical world from a purely algebraic perspective, utilizing ideas from physics such as discrete mass distributions and measures. Learn how multiplication corresponds to traditional convolution of measures and uncover a new kind of symmetry crucial for further development. Topics covered include integral polynumbers, IPoly, integers as coefficients, differences between integral polynumbers and functions, IPoly as measures/densities, rational polynumbers (RPoly), reflection symmetry, and symmetric/alternating integral polynumbers.