# Computational Algorithms for Universal Numbers Rigorous algorithms for computing with universal numbers through their prime-coordinate representations, establishing a comprehensive computational framework aligned with UOR principles. This section presents rigorous algorithms for computing with universal numbers through their prime-coordinate representations, establishing a comprehensive computational framework that aligns with UOR principles. Core representation algorithms include universal number encoding, which translates conventional representations into the universal framework, ensuring consistent encoding across different number domains. The algorithm takes a mathematical object expressed in standard form (integer, rational, real, complex, or p-adic) and outputs the universal number representation including prime coordinates. For integers, the prime factorization algorithm implements the fundamental theorem of arithmetic to generate the prime-coordinate representation. For rational and real numbers, algorithms generate increasingly accurate approximations in prime-coordinate form. For p-adic numbers, algorithms handle the conversion to universal form, ensuring consistency with the p-adic valuation. Basic arithmetic operations on universal numbers include addition, which requires the logarithmic-exponential bridge to translate between additive and multiplicative structures; multiplication, which is directly implemented through coordinate addition, demonstrating the homomorphism property; division, which leverages the homomorphism property to implement division as coordinate subtraction; and exponentiation, which demonstrates how scalar multiplication in coordinate space corresponds to exponentiation of universal numbers. Advanced computational procedures include root finding algorithms that enable finding roots of polynomial and analytic functions in the universal domain; differential equation solvers that solve ODEs in the universal domain, maintaining consistency across all embedded number systems; and Fourier transform algorithms that compute transforms in the universal domain, enabling signal processing applications. Verification and consistency algorithms include zero testing, which is fundamental for computational correctness and requires careful handling of precision; equality testing, which reduces to zero testing of the difference; and precision management algorithms that ensure computational results meet precision requirements across all metrics. Signal processing algorithms implement the universal transform concept and block conversion framework, providing concrete computational procedures that utilize universal coordinates as intermediary representations. Meta-algorithms and framework integration include adaptive precision control, which demonstrates how the universal framework enables resource-efficient computation by adapting precision to task requirements, and the UOR integration framework, which bridges the abstract UOR principles to concrete computational implementations through the universal coordinate system. ## References - [[uor-c-058|representation-algorithms]] - [[uor-c-059|arithmetic-algorithms]] - [[uor-c-060|advanced-computational-procedures]] - [[uor-c-061|verification-algorithms]] ## Metadata - **ID:** urn:uor:resource:computational-algorithms - **Author:** UOR Framework - **Created:** 2025-04-22T00:00:00Z - **Modified:** 2025-04-22T00:00:00Z