# Investigating the Limitations of Transformers with Simple Arithmetic Tasks

arXiv: https://arxiv.org/abs/2102.13019 · HTML: https://mlsanity.com/paper/2102.13019/

## Key facts

- Cited by 139 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 49 references that are in the map.
- First posted on arXiv 25 Feb 2021.

## Builds on (most cited first, 30 of 36)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [Deep contextualized word representations](https://mlsanity.com/paper/1802.05365/) (2018)
- [DeBERTa: Decoding-enhanced BERT with Disentangled Attention](https://mlsanity.com/paper/2006.03654/) (2020)
- [BERT Rediscovers the Classical NLP Pipeline](https://mlsanity.com/paper/1905.05950/) (2019)
- [DROP: A Reading Comprehension Benchmark Requiring Discrete Reasoning Over Paragraphs](https://mlsanity.com/paper/1903.00161/) (2019)
- [Universal Transformers](https://mlsanity.com/paper/1807.03819/) (2018)
- [Scaling Laws for Autoregressive Generative Modeling](https://mlsanity.com/paper/2010.14701/) (2020)
- [Generative Language Modeling for Automated Theorem Proving](https://mlsanity.com/paper/2009.03393/) (2020)
- [Analysing Mathematical Reasoning Abilities of Neural Models](https://mlsanity.com/paper/1904.01557/) (2019)
- [Scaling Laws for Transfer](https://mlsanity.com/paper/2102.01293/) (2021)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [Rethinking Positional Encoding in Language Pre-training](https://mlsanity.com/paper/2006.15595/) (2020)
- [oLMpics -- On what Language Model Pre-training Captures](https://mlsanity.com/paper/1912.13283/) (2019)
- [Do NLP Models Know Numbers? Probing Numeracy in Embeddings](https://mlsanity.com/paper/1909.07940/) (2019)
- [Injecting Numerical Reasoning Skills into Language Models](https://mlsanity.com/paper/2004.04487/) (2020)
- [Neural GPUs Learn Algorithms](https://mlsanity.com/paper/1511.08228/) (2015)
- [Birds have four legs?! NumerSense: Probing Numerical Commonsense Knowledge of Pre-trained Language Models](https://mlsanity.com/paper/2005.00683/) (2020)
- [Inferring Algorithmic Patterns with Stack-Augmented Recurrent Nets](https://mlsanity.com/paper/1503.01007/) (2015)
- [Improve Transformer Models with Better Relative Position Embeddings](https://mlsanity.com/paper/2009.13658/) (2020)
- [Neural Arithmetic Logic Units](https://mlsanity.com/paper/1808.00508/) (2018)
- [Do Language Embeddings Capture Scales?](https://mlsanity.com/paper/2010.05345/) (2020)
- [NumNet: Machine Reading Comprehension with Numerical Reasoning](https://mlsanity.com/paper/1910.06701/) (2019)
- [Encoding word order in complex embeddings](https://mlsanity.com/paper/1912.12333/) (2019)
- [Giving BERT a Calculator: Finding Operations and Arguments with Reading Comprehension](https://mlsanity.com/paper/1909.00109/) (2019)
- [The EOS Decision and Length Extrapolation](https://mlsanity.com/paper/2010.07174/) (2020)
- [EQUATE: A Benchmark Evaluation Framework for Quantitative Reasoning in Natural Language Inference](https://mlsanity.com/paper/1901.03735/) (2019)
- [Enhancing the Transformer with Explicit Relational Encoding for Math Problem Solving](https://mlsanity.com/paper/1910.06611/) (2019)

## Built on by (most cited first, 30 of 139)

- [Train Short, Test Long: Attention with Linear Biases Enables Input Length Extrapolation](https://mlsanity.com/paper/2108.12409/) (2021)
- [PAL: Program-aided Language Models](https://mlsanity.com/paper/2211.10435/) (2022)
- [ToolQA: A Dataset for LLM Question Answering with External Tools](https://mlsanity.com/paper/2306.13304/) (2023)
- [Transformers Learn Shortcuts to Automata](https://mlsanity.com/paper/2210.10749/) (2022)
- [Reasoning with Language Model Prompting: A Survey](https://mlsanity.com/paper/2212.09597/) (2022)
- [Reasoning with Latent Thoughts: On the Power of Looped Transformers](https://mlsanity.com/paper/2502.17416/) (2025)
- [How does GPT-2 compute greater-than?: Interpreting mathematical abilities in a pre-trained language model](https://mlsanity.com/paper/2305.00586/) (2023)
- [What Algorithms can Transformers Learn? A Study in Length Generalization](https://mlsanity.com/paper/2310.16028/) (2023)
- [LIFT: Language-Interfaced Fine-Tuning for Non-Language Machine Learning Tasks](https://mlsanity.com/paper/2206.06565/) (2022)
- [mGPT: Few-Shot Learners Go Multilingual](https://mlsanity.com/paper/2204.07580/) (2022)
- [How well do Large Language Models perform in Arithmetic tasks?](https://mlsanity.com/paper/2304.02015/) (2023)
- [Teaching Arithmetic to Small Transformers](https://mlsanity.com/paper/2307.03381/) (2023)
- [Tokenization counts: the impact of tokenization on arithmetic in frontier LLMs](https://mlsanity.com/paper/2402.14903/) (2024)
- [Teaching Algorithmic Reasoning via In-context Learning](https://mlsanity.com/paper/2211.09066/) (2022)
- [A Primer on the Inner Workings of Transformer-based Language Models](https://mlsanity.com/paper/2405.00208/) (2024)
- [Goat: Fine-tuned LLaMA Outperforms GPT-4 on Arithmetic Tasks](https://mlsanity.com/paper/2305.14201/) (2023)
- [Representing Numbers in NLP: a Survey and a Vision](https://mlsanity.com/paper/2103.13136/) (2021)
- [Exposing Attention Glitches with Flip-Flop Language Modeling](https://mlsanity.com/paper/2306.00946/) (2023)
- [Limitations of Language Models in Arithmetic and Symbolic Induction](https://mlsanity.com/paper/2208.05051/) (2022)
- [GPT Can Solve Mathematical Problems Without a Calculator](https://mlsanity.com/paper/2309.03241/) (2023)
- [Auto-Regressive Next-Token Predictors are Universal Learners](https://mlsanity.com/paper/2309.06979/) (2023)
- [Length Generalization in Arithmetic Transformers](https://mlsanity.com/paper/2306.15400/) (2023)
- [Getting the most out of your tokenizer for pre-training and domain adaptation](https://mlsanity.com/paper/2402.01035/) (2024)
- [Linear algebra with transformers](https://mlsanity.com/paper/2112.01898/) (2021)
- [SuperBPE: Space Travel for Language Models](https://mlsanity.com/paper/2503.13423/) (2025)
- [Positional Description Matters for Transformers Arithmetic](https://mlsanity.com/paper/2311.14737/) (2023)
- [BioT5+: Towards Generalized Biological Understanding with IUPAC Integration and Multi-task Tuning](https://mlsanity.com/paper/2402.17810/) (2024)
- [Interpreting and Improving Large Language Models in Arithmetic Calculation](https://mlsanity.com/paper/2409.01659/) (2024)
- [Pre-trained Large Language Models Use Fourier Features to Compute Addition](https://mlsanity.com/paper/2406.03445/) (2024)
- [Evaluating Large Language Models on Time Series Feature Understanding: A Comprehensive Taxonomy and Benchmark](https://mlsanity.com/paper/2404.16563/) (2024)

## Abstract (arXiv)

The ability to perform arithmetic tasks is a remarkable trait of human intelligence and might form a critical component of more complex reasoning tasks. In this work, we investigate if the surface form of a number has any influence on how sequence-to-sequence language models learn simple arithmetic tasks such as addition and subtraction across a wide range of values. We find that how a number is represented in its surface form has a strong influence on the model's accuracy. In particular, the model fails to learn addition of five-digit numbers when using subwords (e.g., "32"), and it struggles to learn with character-level representations (e.g., "3 2"). By introducing position tokens (e.g., "3 10e1 2"), the model learns to accurately add and subtract numbers up to 60 digits. We conclude that modern pretrained language models can easily learn arithmetic from very few examples, as long as we use the proper surface representation. This result bolsters evidence that subword tokenizers and positional encodings are components in current transformer designs that might need improvement. Moreover, we show that regardless of the number of parameters and training examples, models cannot learn addition rules that are independent of the length of the numbers seen during training. Code to reproduce our experiments is available at https://github.com/castorini/transformers-arithmetic