CNRS UG: Complex Numbers as Single Values: A Working Demonstration of the Complex Numeric Representational System

The Complex Numeric Representational System (CNRS) is a positional numeral system in which each complex number is encoded as a single digit string — analogous to the way the decimal system encodes each real number as a single digit string, without decomposing it into a pair of integers. This paper is the companion guide to the Python module cnrs demo.py, which provides a working, verified implementation of the CNRS arithmetic layer. The paper explains the motivation from first principles, describes each function in the module with worked examples, discusses what the system can and cannot do, and outlines how CNRS arithmetic applies to problems in multi-scale modelling, computational biology, and signal processing where complex-valued quantities arise at multiple scales simultaneously. No prior knowledge of number theory or the CNRS programme is assumed. A reader who is comfortable with complex numbers and basic Python can use this guide to understand, run, and extend the demonstration system.