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Continued fractions are revered in rational approximation theory for their accuracy, precision, and rate of convergence. However, due to the lack of monotonicity in their convergence, continued fractions can be rather difficult to work with. In this talk, we provide a new framework for studying continued fractions utilizing the backwards continued fraction (BCF) which do have monotonic convergence. We show an approximation theory for BCFs, the correspondence between continued fractions and their backwards continued fractions counterpart, and illustrate a rich approximation theory for continued fractions (CFs) utilizing the methods of the approximation theory for the backwards continued fractions. In particular, we construct explicit functions that bound the BCF or CF error over any BCF or CF cylinder set, and along the way work out the details to pass seamlessly between the BCF and CF expansion of any real number. No previous knowledge of continued fractions is necessary. This is joint work with Cameron Bjorklund.