Approximation of Euler’s Number Using Geometry and Probability

Overview

This project explores a novel approach to approximating Euler's Number (e) through fundamental concepts in Geometry and Probability, making it accessible to high school students. Unlike traditional methods that require advanced mathematical knowledge, our approach simplifies the process using basic mathematical principles.

Abstract

Euler's Number (e) is a significant irrational number with numerous applications across various fields. Existing algorithms for approximating e often demand advanced mathematical skills, making them less suitable for high school students. This project introduces a new algorithm for approximating e that leverages high school-level Geometry and Probability.

The methodology is divided into two main components:

1. Theoretical Part: We derive the equation:

   1 / N_u = m * (1 / N_s - 1 / N_e)

   where m = 1, and x represents the value of e. This part focuses on determining areas on a graph to establish a relationship for approximation.

2. Experimental Part: We use random points on the graph to compute areas. Linear Regression is applied to fit a linear equation by adjusting parameters. The experimental results show that with a starting number of points of at least 1,500, and incrementing the number of points by 1,000, a total of 10 points for approximation yields an average Euler's number in the range of 2.5 - 2.7, with a deviation of about 7% from the true value.

Files

• Approx_EulerNumber.ipynb: Contains the experimental code and implementation for approximating Euler's number using the proposed method.

Results

The results indicate that the approximation method can provide a reasonable estimate of Euler's number within the specified conditions. With proper adjustments and sufficient points, the method offers a viable alternative for approximating e using basic high school mathematics.

Getting Started

To run the experiment, ensure you have the required Python environment. Clone this repository and execute the Approx_EulerNumber.ipynb notebook to see the results.

git clone https://github.com/your-username/approximation-euler-number.git
cd approximation-euler-number

License

This project is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. You may not use, copy, modify, or distribute this work in any form without explicit permission from the author.

For more details, see the license.

Acknowledgements

We would like to express our heartfelt gratitude to Manrak Premtong, our advisor for this project, for his invaluable guidance and support. We also extend our thanks to Princess Chulabhorn Science High School Chiang Rai, where this project was developed during our studies.