# What is a Number, That a Large Language Model May Know It?

arXiv: https://arxiv.org/abs/2502.01540 · HTML: https://mlsanity.com/paper/2502.01540/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 52 references that are in the map.
- First posted on arXiv 3 Feb 2025.

## Builds on (most cited first, 28 of 28)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [GPT-4o System Card](https://mlsanity.com/paper/2410.21276/) (2024)
- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [Language Models (Mostly) Know What They Know](https://mlsanity.com/paper/2207.05221/) (2022)
- [Mixtral of Experts](https://mlsanity.com/paper/2401.04088/) (2024)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Probing Classifiers: Promises, Shortcomings, and Advances](https://mlsanity.com/paper/2102.12452/) (2021)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [A Primer in BERTology: What we know about how BERT works](https://mlsanity.com/paper/2002.12327/) (2020)
- [Just Ask for Calibration: Strategies for Eliciting Calibrated Confidence Scores from Language Models Fine-Tuned with Human Feedback](https://mlsanity.com/paper/2305.14975/) (2023)
- [Scaling and evaluating sparse autoencoders](https://mlsanity.com/paper/2406.04093/) (2024)
- [What do you learn from context? Probing for sentence structure in contextualized word representations](https://mlsanity.com/paper/1905.06316/) (2019)
- [Using cognitive psychology to understand GPT-3](https://mlsanity.com/paper/2206.14576/) (2022)
- [Finding Neurons in a Haystack: Case Studies with Sparse Probing](https://mlsanity.com/paper/2305.01610/) (2023)
- [Emergent Linear Representations in World Models of Self-Supervised Sequence Models](https://mlsanity.com/paper/2309.00941/) (2023)
- [Language Models Represent Space and Time](https://mlsanity.com/paper/2310.02207/) (2023)
- [Emergent Analogical Reasoning in Large Language Models](https://mlsanity.com/paper/2212.09196/) (2022)
- [Finding Alignments Between Interpretable Causal Variables and Distributed Neural Representations](https://mlsanity.com/paper/2303.02536/) (2023)
- [How does GPT-2 compute greater-than?: Interpreting mathematical abilities in a pre-trained language model](https://mlsanity.com/paper/2305.00586/) (2023)
- [A Mechanistic Interpretation of Arithmetic Reasoning in Language Models using Causal Mediation Analysis](https://mlsanity.com/paper/2305.15054/) (2023)
- [Measuring Implicit Bias in Explicitly Unbiased Large Language Models](https://mlsanity.com/paper/2402.04105/) (2024)
- [Large language models predict human sensory judgments across six modalities](https://mlsanity.com/paper/2302.01308/) (2023)
- [Time is Encoded in the Weights of Finetuned Language Models](https://mlsanity.com/paper/2312.13401/) (2023)
- [When a language model is optimized for reasoning, does it still show embers of autoregression? An analysis of OpenAI o1](https://mlsanity.com/paper/2410.01792/) (2024)
- [Language Models Encode the Value of Numbers Linearly](https://mlsanity.com/paper/2401.03735/) (2024)
- [The Universal Law of Generalization Holds for Naturalistic Stimuli](https://mlsanity.com/paper/2306.08564/) (2023)

## Built on by (most cited first, 15 of 15)

- [The Lookahead Limitation: Why Multi-Operand Addition is Hard for LLMs](https://mlsanity.com/paper/2502.19981/) (2025)
- [FLAWS: A Benchmark for Error Identification and Localization in Scientific Papers](https://mlsanity.com/paper/2511.21843/) (2025)
- [Levels of Analysis for Large Language Models](https://mlsanity.com/paper/2503.13401/) (2025)
- [Failing to Falsify: Evaluating and Mitigating Confirmation Bias in Language Models](https://mlsanity.com/paper/2604.02485/) (2026)
- [Hypothesis-Driven Feature Manifold Analysis in LLMs via Supervised Multi-Dimensional Scaling](https://mlsanity.com/paper/2510.01025/) (2025)
- [StochasTok: Improving Fine-Grained Subword Understanding in LLMs](https://mlsanity.com/paper/2506.01687/) (2025)
- [Modular Arithmetic: Language Models Solve Math Digit by Digit](https://mlsanity.com/paper/2508.02513/) (2025)
- [The Other Mind: How Language Models Exhibit Human Temporal Cognition](https://mlsanity.com/paper/2507.15851/) (2025)
- [In-Context Algebra](https://mlsanity.com/paper/2512.16902/) (2025)
- [Language Models Do Not Embed Numbers Continuously](https://mlsanity.com/paper/2510.08009/) (2025)
- [Weber's Law in Transformer Magnitude Representations: Efficient Coding, Representational Geometry, and Psychophysical Laws in Language Models](https://mlsanity.com/paper/2603.20642/) (2026)
- [Code-driven Number Sequence Calculation: Enhancing the inductive Reasoning Abilities of Large Language Models](https://mlsanity.com/paper/2510.14620/) (2025)
- [Are Arithmetic Heuristic Neurons Form-Invariant? A Mechanistic Analysis of Symbols, Text, and Code in LLMs](https://mlsanity.com/paper/2607.16693/) (2026)
- [DEL: Digit Entropy Loss for Numerical Learning of Large Language Models](https://mlsanity.com/paper/2605.20369/) (2026)
- [Contextual trajectory and incremental contextual displacement: Towards using LLMs to understand dynamic, utterance-specific meaning construction](https://mlsanity.com/paper/2610.00840/) (2026)

## Abstract (arXiv)

Numbers are a basic part of how humans represent and describe the world around them. As a consequence, learning effective representations of numbers is critical for the success of large language models as they become more integrated into everyday decisions. However, these models face a challenge: depending on context, the same sequence of digit tokens, e.g., 911, can be treated as a number or as a string. What kind of representations arise from this duality, and what are its downstream implications? Using a similarity-based prompting technique from cognitive science, we show that LLMs learn representational spaces that blend string-like and numerical representations. In particular, we show that elicited similarity judgments from these models over integer pairs can be captured by a combination of Levenshtein edit distance and numerical Log-Linear distance, suggesting an entangled representation. In a series of experiments we show how this entanglement is reflected in the latent embeddings, how it can be reduced but not entirely eliminated by context, and how it can propagate into a realistic decision scenario. These results shed light on a representational tension in transformer models that must learn what a number is from text input.