# Unraveling Arithmetic in Large Language Models: The Role of Algebraic Structures

arXiv: https://arxiv.org/abs/2411.16260 · HTML: https://mlsanity.com/paper/2411.16260/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 25 of its 32 references that are in the map.
- First posted on arXiv 25 Nov 2024.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 25 of 25)

- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Show Your Work: Scratchpads for Intermediate Computation with Language Models](https://mlsanity.com/paper/2112.00114/) (2021)
- [Towards Revealing the Mystery behind Chain of Thought: A Theoretical Perspective](https://mlsanity.com/paper/2305.15408/) (2023)
- [How does GPT-2 compute greater-than?: Interpreting mathematical abilities in a pre-trained language model](https://mlsanity.com/paper/2305.00586/) (2023)
- [MathScale: Scaling Instruction Tuning for Mathematical Reasoning](https://mlsanity.com/paper/2403.02884/) (2024)
- [Why think step by step? Reasoning emerges from the locality of experience](https://mlsanity.com/paper/2304.03843/) (2023)
- [A Mechanistic Interpretation of Arithmetic Reasoning in Language Models using Causal Mediation Analysis](https://mlsanity.com/paper/2305.15054/) (2023)
- [Goat: Fine-tuned LLaMA Outperforms GPT-4 on Arithmetic Tasks](https://mlsanity.com/paper/2305.14201/) (2023)
- [Auto-Regressive Next-Token Predictors are Universal Learners](https://mlsanity.com/paper/2309.06979/) (2023)
- [Improving Large Language Model Fine-tuning for Solving Math Problems](https://mlsanity.com/paper/2310.10047/) (2023)
- [Interpreting and Improving Large Language Models in Arithmetic Calculation](https://mlsanity.com/paper/2409.01659/) (2024)
- [Language Models Encode Numbers Using Digit Representations in Base 10](https://mlsanity.com/paper/2410.11781/) (2024)
- [NumeroLogic: Number Encoding for Enhanced LLMs' Numerical Reasoning](https://mlsanity.com/paper/2404.00459/) (2024)
- [Latent Space Symmetry Discovery](https://mlsanity.com/paper/2310.00105/) (2023)
- [A Theory for Length Generalization in Learning to Reason](https://mlsanity.com/paper/2404.00560/) (2024)
- [Language Models are Symbolic Learners in Arithmetic](https://mlsanity.com/paper/2410.15580/) (2024)
- [Dual Instruction Tuning with Large Language Models for Mathematical Reasoning](https://mlsanity.com/paper/2403.18295/) (2024)
- [A Unified Framework for Discovering Discrete Symmetries](https://mlsanity.com/paper/2309.02898/) (2023)
- [Adapting While Learning: Grounding LLMs for Scientific Problems with Intelligent Tool Usage Adaptation](https://mlsanity.com/paper/2411.00412/) (2024)
- [RevOrder: A Novel Method for Enhanced Arithmetic in Language Models](https://mlsanity.com/paper/2402.03822/) (2024)
- [Executing Arithmetic: Fine-Tuning Large Language Models as Turing Machines](https://mlsanity.com/paper/2410.07896/) (2024)
- [Neural Discovery of Permutation Subgroups](https://mlsanity.com/paper/2309.05352/) (2023)
- [Exploring Group and Symmetry Principles in Large Language Models](https://mlsanity.com/paper/2402.06120/) (2024)
- [Learning Mathematical Rules with Large Language Models](https://mlsanity.com/paper/2410.16973/) (2024)

## Built on by (most cited first, 4 of 4)

- [Do Large Language Models Truly Grasp Addition? A Rule-Focused Diagnostic Using Two-Integer Arithmetic](https://mlsanity.com/paper/2504.05262/) (2025)
- [Safety Alignment Depth in Large Language Models: A Markov Chain Perspective](https://mlsanity.com/paper/2502.00669/) (2025)
- [The Shape of Addition: Geometric Structures of Arithmetic in Large Language Models](https://mlsanity.com/paper/2606.03645/) (2026)
- [Enhancing Transformer Representations of Symbolic ODE Expressions](https://mlsanity.com/paper/2609.24746/) (2026)

## Abstract (arXiv)

Large language models (LLMs) have demonstrated remarkable mathematical capabilities, largely driven by chain-of-thought (CoT) prompting, which decomposes complex reasoning into step-by-step solutions. This approach has enabled significant advancements, as evidenced by performance on benchmarks like GSM8K and MATH. However, the mechanisms underlying LLMs' ability to perform arithmetic in a single step of CoT remain poorly understood. Existing studies debate whether LLMs encode numerical values or rely on symbolic reasoning, while others explore attention and multi-layered processing in arithmetic tasks. In this work, we propose that LLMs learn arithmetic by capturing algebraic structures, such as commutativity and identity properties. Since these structures are observable through input-output relationships, they can generalize to unseen data. We empirically demonstrate that LLMs can learn algebraic structures using a custom dataset of arithmetic problems, as well as providing theoretical evidence showing that, under specific configurations of weights and biases, the transformer-based LLMs can generate embeddings that remain invariant to both permutations of input tokens and the presence of identity elements. Our findings indicate that leveraging algebraic structures can enhance the LLMs' arithmetic capabilities, offering insights into improving their arithmetic performance.