Dual complex numbers and Leibniz's differentiation rules
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We are aiming to explain a purely algebraic approach to infinitesimals that extends differential calculus to general fields -- even to finite fields. The dual complex numbers are another commutative subalgebra of the algebra of Dihedrons. They were introduced by William Clifford in 1873. To set up this theory, we first have to review the data structure approach to polynomials, called polynumbers. These are not functions, but rather expressions defined over a general field, which can be evaluated at elements of the field. Happily they can also be evaluated at dual numbers over that field, which include the infinitesimal "epsilon". This is not some "infinitely small quantity", but just a special, and rather simple 2 x 2 matrix. We also introduce Leibniz's rules of differentiation that we are aiming to understand, and that go back before Leibniz to the first person to introduce the Derivative --Johann Faulhaber. A correction to pg 4, second line: the condition is for l greater than k, not for l greater than 0. ************************ Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/...My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated