LLM-Generated Numerical Representations

• LLM-generated numerical representations are high-dimensional encodings combining digit-level, linear, and compressed patterns to support arithmetic and comparative tasks.
• They integrate numeric reasoning and language processing through internal embeddings that mimic a mental number line, yet struggle with abstraction and generalization.
• Engineering strategies such as prompt design, domain tuning, and code synthesis help enhance numerical robustness and mitigate intrinsic limitations.

LLM-generated numerical representations refer to the internal encoding, manipulation, and utilization of numbers and numeric concepts within transformer-based LLMs. Unlike symbolic computation systems, LLMs learn to represent numbers as patterns in high-dimensional vector spaces through language-driven statistical learning. These representations underpin the models’ ability to perform numerical reasoning, arithmetic, numeric comparisons, and integrate quantitative information into language understanding tasks. Recent research elucidates both the structural properties and behavioral outcomes of LLMs’ numerical processing, highlighting parallels and divergences with human cognition, as well as exposing limitations in abstraction, generalization, and robustness.

1. Structure and Geometry of Numerical Representations

LLMs do not encode numbers exclusively as single, holistic magnitudes; instead, their representations can be multifaceted, blending digit-level, string-based, and value-based components:

• Digit-wise Circular Representations: Internal activations decompose numbers into digit slots, with each digit $t$ in position $i$ represented as a point on a base-$b$ unit circle: $[\cos(2\pi t/b), \sin(2\pi t/b)]$, typically with $b=10$ (Levy et al., 15 Oct 2024). Probing experiments confirm that these circular encodings facilitate per-digit decoding and causally determine outputs in arithmetic tasks.
• Linear Value Subspaces: Projection analyses show numerical information is predominantly localized in low-dimensional linear subspaces of the high-dimensional embedding space. Partial Least Squares (PLS) projections reveal that attributes such as years or latitudes are encoded along a single or few directions, enabling direct linear extraction and manipulation for comparison tasks (El-Shangiti et al., 17 Oct 2024).
• Logarithmic and Sublinear Compression: Principal Component Analysis (PCA) and geometric regression across number embeddings indicate that LLMs, like humans, compress numerical magnitudes sublinearly—distances between larger values contract, forming a mental number line akin to logarithmic scaling (AlquBoj et al., 22 Feb 2025, Shah et al., 2023). The function $f_{LLM}(x)\approx \alpha\beta^i$ (with $\beta<1$) fits numerical embeddings for $x=10^i$, mirroring cognitive phenomena in human psychophysics.
• Entangled String–Number Representations: LLM embeddings blend numerical magnitude with string similarity, as measured by a linear combination of log-linear (magnitude) distance and Levenshtein (edit) distance. Context can modulate but not fully disentangle these components (Marjieh et al., 3 Feb 2025).

2. Behavioral Benchmarks and Cognitive Parallels

When evaluated through behavioral benchmarks inspired by cognitive science:

• Distance, Size, and Ratio Effects: The "distance effect" (greater discriminability with greater magnitude difference), "size effect" (smaller numbers are more easily compared), and "ratio effect" (comparison difficulty scales with numerical ratio) all manifest in LLM internal similarity metrics, with high $R^2$ values for digit inputs (Shah et al., 2023).
• Mental Number Line Emergence: Multidimensional scaling applied to LLM distance matrices recovers a compressed, approximately log-spaced "mental number line," again resembling human magnitude representation, particularly in models like GPT-2 when the input forms are digits (Shah et al., 2023).
• Non-literal Understanding Shortcomings: Unlike humans, LLMs default to literal interpretations of number words, showing deficiencies in understanding pragmatic halo effects (imprecise meaning of round numbers) and hyperbole. These differences are attributed not to the lack of world knowledge priors, but to the manner logical inference is performed with these priors (Tsvilodub et al., 10 Feb 2025).

3. Mechanisms for Numerical Reasoning and Calculation

• Linear Encoding and Probing: Linear probes can effectively read out the approximate numerical value from hidden activations (typically with a log or log2 transform), achieving high correlation but limited exact accuracy, especially in intermediate layers where representations are closest to linear (Zhu et al., 8 Jan 2024). Intervention experiments with hidden state vector addition can causally perturb outputs, linking internal representation directly to model decisions.
• Symbolic (Discrete State) Representations: Models develop "implicit discrete state representations" (IDSRs) at key token or layer positions, allowing multi-step arithmetic (e.g., chained additions) to be accomplished without explicit chain-of-thought reasoning (Chen et al., 16 Jul 2024). Probing reveals these states are not lossless—representation fidelity degrades across sequential computation.
• Failure Modes in Abstraction: Despite high accuracy on standard numerical tasks, LLMs struggle to generalize learned rules under abstraction, as shown by dramatic accuracy drops (to ≤7.5%) when addition tasks are symbolically re-encoded with bijective mappings (e.g., digit-to-symbol). This indicates reliance on memorized surface patterns rather than true algorithmic rule learning, as further supported by failures in commutativity and compositional generalization (Yan et al., 7 Apr 2025).

4. Practical Strategies and Engineering Interventions

• Prompt and Representation Engineering: Explicitly augmenting number representations—such as prefixing counts of digits (NumeroLogic encoding)—substantially improves numerical reasoning in arithmetic tasks and general benchmarks, serving as an implicit "Chain of Thought" mechanism (Schwartz et al., 30 Mar 2024).
• Input Format Alignment: Reformatting numerical sequences into code-like structures (e.g., Python lists, dictionaries) enhances performance in data-to-text tasks compared to plain or verbose formats, reflecting greater alignment with LLM pretraining distributions (Kawarada et al., 3 Apr 2024).
• Domain Adaptation and Tuning: Numeric-sensitive models for finance (e.g., NumLLM) employ curated financial corpora and dual LoRA modules for continual pre-training and numeric-specific tuning, then merge them to achieve top accuracy on domain-specific QA benchmarks, especially for numeric variables (Su et al., 1 May 2024). Instruction tuning with rich tag metadata and parameter-efficient LoRA further allows generative numeric annotation in financial settings, achieving superior zero-shot and rare-label performance (Khatuya et al., 3 May 2024).
• Program Generation for Differential Testing: LLM-guided code synthesis frameworks (LLM4FP) leverage LLMs to generate floating-point programs that systematically expose numerical inconsistencies across compilers, surpassing traditional fuzzers in discovering "Real vs. Real" output differences, not just exceptional cases (NaN, Inf) (Wang et al., 29 Aug 2025).

5. Limitations, Fragility, and Failure Cases

• Number Sense Fragility: Despite superficially strong performance on deterministic arithmetic, LLMs exhibit a brittle "number sense"—they underperform in tasks demanding combinatorial or trial-and-error reasoning (e.g., the Game of 24), with accuracy falling from >90% to as low as 10-27% as the complexity or search bottleneck increases (Rahman, 31 Mar 2025, Rahman et al., 8 Sep 2025).
• Overreliance on Pattern Matching: Error audits reveal that LLMs primarily recall procedural patterns rather than generalize to new, out-of-distribution or symbolically abstracted settings. Non-monotonic scaling with input size and frequent algebraic property violations (e.g., commutativity) further support this conclusion (Yan et al., 7 Apr 2025).
• Contextual and Linguistic Confounds: Entanglement of number and string representations causes models to select options closer in string form rather than numeric value in ambiguous tasks, especially for longer numerals (Marjieh et al., 3 Feb 2025). Translation tasks involving units, large values, or fractional conversions reveal persistent mistranslation issues; post-editing pipelines—based on extraction and arithmetic verification—are necessitated to ensure correctness (Tang et al., 9 Jan 2025).

6. Implications and Future Directions

• Interpretability and Mechanistic Analysis: Understanding the geometric, sublinear, and digit-wise nature of numeric representations informs both cognitive modeling and the mechanistic interpretability of LLMs (AlquBoj et al., 22 Feb 2025, Levy et al., 15 Oct 2024, Shah et al., 2023).
• Model Editing and Control: The uncovering of linear subspaces and causal interventions in those subspaces suggests new avenues for debugging, editing, or steering model behavior in quantitative tasks (El-Shangiti et al., 17 Oct 2024).
• Enhancing Numerical Robustness: Recommendations include: explicit multi-context training, hybrid symbolic-connectionist architectures, improved error-correction modules operating on digit-wise or value-based slots, and enhanced search or planning modules (e.g., Tree of Thoughts) to overcome bottlenecks in combinatorial search (Rahman, 31 Mar 2025, Rahman et al., 8 Sep 2025).
• Evaluation and Benchmarking: The data underscore the necessity for targeted, simple tests of low-level numerical reasoning (e.g., Numberland-style, symbolic generalization probes) and deliberate stress-testing on both deterministic and uncertain tasks, as high aggregate benchmark performance can mask fundamental deficits (Rahman, 31 Mar 2025, Yan et al., 7 Apr 2025).

7. Conclusions

LLM-generated numerical representations possess a hybrid, context-dependent structure that supports a range of quantitative linguistic tasks via compressed, sublinear geometric patterns, linear attribute directions, digit slot encodings, and partial cognitive analogs to human number sense. Despite behavioral effects mirroring aspects of human cognition, they remain limited by entangled representations, lack of true abstraction, and fragility in combinatorial reasoning. Progress in architecture, training, and evaluation is needed to advance beyond statistically learned surface patterns toward robust, compositional, and truly rule-based numerical intelligence in LLMs.