# GPT Can Solve Mathematical Problems Without a Calculator

arXiv: https://arxiv.org/abs/2309.03241 · HTML: https://mlsanity.com/paper/2309.03241/

## Key facts

- Cited by 68 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 45 references that are in the map.
- First posted on arXiv 6 Sep 2023.

## Builds on (most cited first, 30 of 40)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [Large Language Models are Zero-Shot Reasoners](https://mlsanity.com/paper/2205.11916/) (2022)
- [Toolformer: Language Models Can Teach Themselves to Use Tools](https://mlsanity.com/paper/2302.04761/) (2023)
- [OPT: Open Pre-trained Transformer Language Models](https://mlsanity.com/paper/2205.01068/) (2022)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Beyond the Imitation Game: Quantifying and extrapolating the capabilities of language models](https://mlsanity.com/paper/2206.04615/) (2022)
- [BLOOM: A 176B-Parameter Open-Access Multilingual Language Model](https://mlsanity.com/paper/2211.05100/) (2022)
- [Least-to-Most Prompting Enables Complex Reasoning in Large Language Models](https://mlsanity.com/paper/2205.10625/) (2022)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [LaMDA: Language Models for Dialog Applications](https://mlsanity.com/paper/2201.08239/) (2022)
- [Scaling Language Models: Methods, Analysis & Insights from Training Gopher](https://mlsanity.com/paper/2112.11446/) (2021)
- [Program of Thoughts Prompting: Disentangling Computation from Reasoning for Numerical Reasoning Tasks](https://mlsanity.com/paper/2211.12588/) (2022)
- [GLM: General Language Model Pretraining with Autoregressive Blank Infilling](https://mlsanity.com/paper/2103.10360/) (2021)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [Show Your Work: Scratchpads for Intermediate Computation with Language Models](https://mlsanity.com/paper/2112.00114/) (2021)
- [GLM-130B: An Open Bilingual Pre-trained Model](https://mlsanity.com/paper/2210.02414/) (2022)
- [Galactica: A Large Language Model for Science](https://mlsanity.com/paper/2211.09085/) (2022)
- [PAL: Program-aided Language Models](https://mlsanity.com/paper/2211.10435/) (2022)
- [Using DeepSpeed and Megatron to Train Megatron-Turing NLG 530B, A Large-Scale Generative Language Model](https://mlsanity.com/paper/2201.11990/) (2022)
- [CogView2: Faster and Better Text-to-Image Generation via Hierarchical Transformers](https://mlsanity.com/paper/2204.14217/) (2022)

## Built on by (most cited first, 30 of 68)

- [MAmmoTH: Building Math Generalist Models through Hybrid Instruction Tuning](https://mlsanity.com/paper/2309.05653/) (2023)
- [Large Language Models for Mathematical Reasoning: Progresses and Challenges](https://mlsanity.com/paper/2402.00157/) (2024)
- [From Explicit CoT to Implicit CoT: Learning to Internalize CoT Step by Step](https://mlsanity.com/paper/2405.14838/) (2024)
- [InternLM-Math: Open Math Large Language Models Toward Verifiable Reasoning](https://mlsanity.com/paper/2402.06332/) (2024)
- [MARIO: MAth Reasoning with code Interpreter Output -- A Reproducible Pipeline](https://mlsanity.com/paper/2401.08190/) (2024)
- [Transformers Can Do Arithmetic with the Right Embeddings](https://mlsanity.com/paper/2405.17399/) (2024)
- [GraphInstruct: Empowering Large Language Models with Graph Understanding and Reasoning Capability](https://mlsanity.com/paper/2403.04483/) (2024)
- [Positional Description Matters for Transformers Arithmetic](https://mlsanity.com/paper/2311.14737/) (2023)
- [A Survey of Mathematical Reasoning in the Era of Multimodal Large Language Model: Benchmark, Method & Challenges](https://mlsanity.com/paper/2412.11936/) (2024)
- [MathCoder-VL: Bridging Vision and Code for Enhanced Multimodal Mathematical Reasoning](https://mlsanity.com/paper/2505.10557/) (2025)
- [MathCoder2: Better Math Reasoning from Continued Pretraining on Model-translated Mathematical Code](https://mlsanity.com/paper/2410.08196/) (2024)
- [Step-Controlled DPO: Leveraging Stepwise Error for Enhanced Mathematical Reasoning](https://mlsanity.com/paper/2407.00782/) (2024)
- [Position: Multimodal Large Language Models Can Significantly Advance Scientific Reasoning](https://mlsanity.com/paper/2502.02871/) (2025)
- [AutoGeo: Automating Geometric Image Dataset Creation for Enhanced Geometry Understanding](https://mlsanity.com/paper/2409.09039/) (2024)
- [Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers](https://mlsanity.com/paper/2502.02393/) (2025)
- [How Far Have We Gone in Vulnerability Detection Using Large Language Models](https://mlsanity.com/paper/2311.12420/) (2023)
- [Mathematical Language Models: A Survey](https://mlsanity.com/paper/2312.07622/) (2023)
- [Materials science in the era of large language models: a perspective](https://mlsanity.com/paper/2403.06949/) (2024)
- [Mathematical Reasoning in Large Language Models: Assessing Logical and Arithmetic Errors across Wide Numerical Ranges](https://mlsanity.com/paper/2502.08680/) (2025)
- [MathGLM-Vision: Solving Mathematical Problems with Multi-Modal Large Language Model](https://mlsanity.com/paper/2409.13729/) (2024)
- [Large Language Model Reasoning Failures](https://mlsanity.com/paper/2602.06176/) (2026)
- [An overview of domain-specific foundation model: key technologies, applications and challenges](https://mlsanity.com/paper/2409.04267/) (2024)
- [M2-omni: Advancing Omni-MLLM for Comprehensive Modality Support with Competitive Performance](https://mlsanity.com/paper/2502.18778/) (2025)
- [BAM! Just Like That: Simple and Efficient Parameter Upcycling for Mixture of Experts](https://mlsanity.com/paper/2408.08274/) (2024)
- [Regression-aware Inference with LLMs](https://mlsanity.com/paper/2403.04182/) (2024)
- [Adapting Large Language Models for Education: Foundational Capabilities, Potentials, and Challenges](https://mlsanity.com/paper/2401.08664/) (2023)
- [Tele-FLM Technical Report](https://mlsanity.com/paper/2404.16645/) (2024)
- [A Survey on Mathematical Reasoning and Optimization with Large Language Models](https://mlsanity.com/paper/2503.17726/) (2025)
- [Explicitly Encoding Structural Symmetry is Key to Length Generalization in Arithmetic Tasks](https://mlsanity.com/paper/2406.01895/) (2024)
- [Language Models are Symbolic Learners in Arithmetic](https://mlsanity.com/paper/2410.15580/) (2024)

## Abstract (arXiv)

Previous studies have typically assumed that large language models are unable to accurately perform arithmetic operations, particularly multiplication of >8 digits, and operations involving decimals and fractions, without the use of calculator tools. This paper aims to challenge this misconception. With sufficient training data, a 2 billion-parameter language model can accurately perform multi-digit arithmetic operations with almost 100% accuracy without data leakage, significantly surpassing GPT-4 (whose multi-digit multiplication accuracy is only 4.3%). We also demonstrate that our MathGLM, fine-tuned from GLM-10B on a dataset with additional multi-step arithmetic operations and math problems described in text, achieves similar performance to GPT-4 on a 5,000-samples Chinese math problem test set. Our code and data are public at https://github.com/THUDM/MathGLM.