We continue our exploration of differentials as arbitrarily small real numbers and develop various algebraic structures including semigroup algebras and the dual algebra of differentials and inverse differentials as a mapping onto the non-zero real numbers. We also discuss the notion of families of neighborhoods as fibers in a fiber bundle of differentials on the real number line. We explore the subject in an intuitive fashion, eventually leading to a rigorous theory of differentials as arbitrarily small numbers in epsilon neighborhoods of a given real number.
