<![CDATA[This research introduces a novel set of mathematical formulae for efficiently determining the frequency of individual digits within the set of natural numbers less than a given integer N. The study aims to derive a general closed-form expression that avoids iteration, based on the digit structure of N, and to develop a new positional operator for identifying digit placement within multi-digit numbers. The methodology is built on place-value analysis and the definition of a digit frequency function f(t;N), which incorporates a universal positional term fn and a helper function G(ai,t). The formula f(t;N) = fn + ∑G(ai,t) is proven to hold across numerical classes and is validated through extensive numerical testing up to 10^18. The research also introduces a novel mathematical operator, Ta = t-1, to determine the exact placement of a digit at the end of a number sequence. The results demonstrate a 92% improvement in computational efficiency compared to enumeration, with broad applications in number theory, coding, and pattern recognition. Additionally, the approach resolves a known non-linear radical system with exact solutions, showcasing the formulae's algebraic utility. In conclusion, this study contributes three new tools to mathematics: a closed-form digit frequency function, a terminal digit positional operator, and a novel solution method for radical equations.]]>
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