We begin moving towards calculus with polynumbers/polynomials by introducing the Derivative D=D_1 in a simple algebraic way. First we discuss composition of integral polynumbers, and the translation of a polynumber by an integer. This leads to the Taylor (bi) polynumber/polynomial of a polynumber/polynomial, which contains not only the usual derivative, but also higher analogs called sub-derivatives, denoted D_2, D_3 etc. This lecture should be disorienting but exciting to those who have been indoctrinated into thinking that calculus is analysis, resting on limit notions and so called `real numbers'. A dramatical shift of the mathematical landscape is at our doorstep: a shift going back to Euler and Lagrange.This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. The full playlist is at http://www.youtube.com/playlist?list=PL5A714C94D40392AB&feature=view_all
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