Explore the concept of Boole polynumbers and their relationship to logic circuits in this 38-minute mathematics lecture. Delve into the distinction between polynumbers and polynomials, focusing on coefficient arrays rather than variables. Examine how coefficients from the bifield B_2={0,1} lead to Poly B_2, and discover how Boole polynumbers serve as natural representations of Boolean functions and logic gate effects. Learn about the importance of Boole polynumbers as invariants for equivalent circuits. Investigate an example from Mandy Orzechowski's work, and gain insights into polynumber evaluation, non-functional aspects of polynumbers, and the fundamental theorem of Boole polynumber evaluation.