This paper introduces the Adaptive Base Representation (ABR) Theorem and proposes a novel number system that offers a structured alternative to the binary number system for digital computers. The ABR number system enables each decimal number to be represented uniquely and using the same number of bits, $n$, as the binary encoding. Theoretical foundations and mathematical formulations demonstrate that ABR can encode the same integer range as binary, validating its potential as a viable alternative. Additionally, the ABR number system is compatible with existing data compression algorithms like Huffman coding and arithmetic coding, as well as error detection and correction mechanisms such as Hamming codes. We further explore practical applications, including digital steganography, to illustrate the utility of ABR in information theory and digital encoding, suggesting that the ABR number system could inspire new approaches in digital data representation and computational design.

翻译：本文介绍了自适应基数表示定理，并提出了一种新颖的数制，为数字计算机提供了一种结构化的二进制数制替代方案。ABR数制使得每个十进制数能够以与二进制编码相同的比特数$n$进行唯一表示。理论基础和数学公式表明，ABR能够编码与二进制相同的整数范围，验证了其作为一种可行替代方案的潜力。此外，ABR数制与现有的数据压缩算法（如霍夫曼编码和算术编码）以及错误检测与纠正机制（如汉明码）兼容。我们进一步探讨了实际应用，包括数字隐写术，以说明ABR在信息论和数字编码中的效用，表明ABR数制可能为数字数据表示和计算设计领域带来新的方法。