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 × 10² 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.

Federated Learning (FL) enables privacy-preserving collaborative instruction tuning of large language models (LLMs) by leveraging massively distributed data. However, the decentralized nature of FL exacerbates data quality challenges, as local clients lack global visibility to filter noisy or low-quality samples before training. To resolve this issue, we propose FedDQC, a novel federated instruction tuning framework with dynamic data quality control. Our approach introduces two key innovations. First, we propose instruction-response alignment (IRA)—an efficient client-side metric for quality evaluation requiring only low-cost inference. We validate that higher-IRA data corresponds to more relevant and easier-to-learn question-answer pairs. Second, mirroring the human easy-to-hard knowledge acquisition process, we design a quality-aware hierarchical FL training framework, where the LLM is progressively fine-tuned from high- to low-IRA data in a collaborative manner. The framework also supports adaptive data quality assessment at each hierarchy, enabling dynamic adjustments throughout the training process. Extensive experiments on synthetic and real-world datasets show that our method significantly improves LLM performance on mixed-quality data in FL.