# Visual Learning of Arithmetic Operations

arXiv: https://arxiv.org/abs/1506.02264 · HTML: https://mlsanity.com/paper/1506.02264/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2015.

## Built on by (most cited first, 3 of 3)

- [Towards Understanding Grokking: An Effective Theory of Representation Learning](https://mlsanity.com/paper/2205.10343/) (2022)
- [The Clock and the Pizza: Two Stories in Mechanistic Explanation of Neural Networks](https://mlsanity.com/paper/2306.17844/) (2023)
- [Acceleration of Grokking in Learning Arithmetic Operations via Kolmogorov-Arnold Representation](https://mlsanity.com/paper/2405.16658/) (2024)

## Abstract (arXiv)

A simple Neural Network model is presented for end-to-end visual learning of arithmetic operations from pictures of numbers. The input consists of two pictures, each showing a 7-digit number. The output, also a picture, displays the number showing the result of an arithmetic operation (e.g., addition or subtraction) on the two input numbers. The concepts of a number, or of an operator, are not explicitly introduced. This indicates that addition is a simple cognitive task, which can be learned visually using a very small number of neurons. Other operations, e.g., multiplication, were not learnable using this architecture. Some tasks were not learnable end-to-end (e.g., addition with Roman numerals), but were easily learnable once broken into two separate sub-tasks: a perceptual textitCharacter Recognition and cognitive textitArithmetic sub-tasks. This indicates that while some tasks may be easily learnable end-to-end, other may need to be broken into sub-tasks.