LLMs Know More About Numbers than They Can Say

2026-02-08

AI Summary

This study addresses the poor performance of large language models (LLMs) in numerical comparison tasks involving mixed notations (e.g., “5.7×10² vs. 580”), where they struggle to produce accurate explicit reasoning. We reveal for the first time that LLMs internally encode magnitude and relative size information in their hidden representations, yet fail to effectively leverage this knowledge in their outputs. To investigate, we employ linear probing to analyze internal representations and introduce a combined approach using magnitude regression and ranking classification. Our method achieves relative errors of 2.3% on synthetic data and 19.06% on scientific papers, with ranking accuracy exceeding 90%. Furthermore, incorporating the probe loss as an auxiliary objective during fine-tuning improves explicit numerical reasoning accuracy by 3.22%.

Technology Category

• Machine Learning: Large Multimodal Models (LMMs)
• Natural Language Processing: (Large) Language Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Although state-of-the-art LLMs can solve math problems, we find that they make errors on numerical comparisons with mixed notation:"Which is larger, $5.7 \times 10^2$ or $580$?"This raises a fundamental question: Do LLMs even know how big these numbers are? We probe the hidden states of several smaller open-source LLMs. A single linear projection of an appropriate hidden layer encodes the log-magnitudes of both kinds of numerals, allowing us to recover the numbers with relative error of about 2.3% (on restricted synthetic text) or 19.06% (on scientific papers). Furthermore, the hidden state after reading a pair of numerals encodes their ranking, with a linear classifier achieving over 90% accuracy. Yet surprisingly, when explicitly asked to rank the same pairs of numerals, these LLMs achieve only 50-70% accuracy, with worse performance for models whose probes are less effective. Finally, we show that incorporating the classifier probe's log-loss as an auxiliary objective during finetuning brings an additional 3.22% improvement in verbalized accuracy over base models, demonstrating that improving models'internal magnitude representations can enhance their numerical reasoning capabilities.

Problem

• numerical reasoning
• large language models
• number comparison
• magnitude representation
• mixed notation

Innovation

• numerical reasoning
• hidden state probing
• log-magnitude representation
• auxiliary fine-tuning
• linear probe