Treffer: A Non-Iterative Method for Reciprocal Computation by Exact Error Cancellation

We introduce a novel, non-iterative algorithm for the exact determination of the reciprocalof any integer, n. The method is based on the finding that a single integer division, 10k ÷ n(where k is a function of the number of digits in n), provides all the necessary componentsfor the calculation. The quotient of this division, N1, is used to form an initial estimate, a,while the remainder, R, is used to construct a perfect correction factor, r. We demonstratethat applying these components within the geometric series summation formula, s = a/(1 − r),the series algebraically collapses to the exact identity 1/n. The algorithm’s core is the proofthat the remainder R is equivalent to the integer ratio between the first two digit blocks of thereciprocal’s decimal expansion, thus justifying this direct computational approach. This methodre-contextualizes a tool of infinite series approximation as a finite, direct procedure for exactrational arithmetic.