<p>The mathematical foundations of data encoding and performing modular operations in the residue number system (RNS) are presented. The mathematical transformations of the system of orthogonal harmonic functions of the Fourier number-theoretic basis are investigated. Models for the formation of RNS codes based on phase portraits of harmonic functions whose frequencies meet the conditions of being coprime are constructed. Models of the formation of discrete-quantum sawtooth functions of the phase portraits of harmonic frequencies in the system of residue number modules <i>P</i>₁ = 2, <i>P</i>₂ = 3,<i> P</i>₃ = 5 are considered. Algorithms for the formation and processing of digital data represented by codes of the smallest non-negative residues are presented. Algorithms for performing computing operations in the codes of integer, normalized, and perfect forms of RNS are analyzed. Methods and algorithms for performing high-performance arithmetic and logical operations in RNS codes are proposed. The performance characteristics of arithmetic and logical operations in RNS of digital data represented in the Rademacher, Rademacher–Krestenson, and Haar–Krestenson number-theoretic bases are analyzed. Algorithms for comparing numbers in RNS codes are analyzed. Methods for forming residue codes based on analog and digital data are considered. A method for converting binary codes of the Rademacher number-theoretic basis into residue codes RNS modulo is developed. An algorithm is constructed to determine the sample mathematical expectation in RNS by calculating the sum of the ranks of stream data processing. A method for converting multi-digit numbers represented in RNS into binary codes of the Rademacher number-theoretic basis is analyzed. The structures of special-purpose processors that implement computing tasks in RNS are presented.</p>