Large language models (LLMs) have exhibited impressive competence in various tasks, but their internal mechanisms on mathematical problems are still under-explored. In this paper, we study a fundamental question: how language models encode the value of numbers, a basic element in math. To study the question, we construct a synthetic dataset comprising addition problems and utilize linear probes to read out input numbers from the hidden states. Experimental results support the existence of encoded number values in LLMs on different layers, and these values can be extracted via linear probes. Further experiments show that LLMs store their calculation results in a similar manner, and we can intervene the output via simple vector additions, proving the causal connection between encoded numbers and language model outputs. Our research provides evidence that LLMs encode the value of numbers linearly, offering insights for better exploring, designing, and utilizing numeric information in LLMs.

翻译：大型语言模型（LLM）在各种任务中展现出卓越的能力，但其处理数学问题的内部机制仍有待深入探索。本文研究一个基础性问题：语言模型如何编码数字这一数学基本元素的值。为探究此问题，我们构建了一个包含加法运算的合成数据集，并利用线性探针从隐藏状态中读取输入数字。实验结果表明，LLM在不同层中确实存在编码的数字值，且这些值可通过线性探针提取。进一步实验显示，LLM以类似方式存储计算结果，并能通过简单的向量加法干预输出，从而证明编码数字与语言模型输出之间存在因果关联。本研究为LLM线性编码数字值提供了证据，为更好地探索、设计和利用LLM中的数值信息提供了见解。