Exploring University Mathematics with Python

Description

A typical mathematics university curriculum would include some element of programming, usually in a standalone module. However, in my experience teaching at several UK universities, students often regard programming as just another module that is disparate from a typical ‘pen-and-paper’ module. In my opinion, this is an extremely unhealthy viewpoint, because programming can often help us gain a more solid understanding of mathematics in comparison to a purely pen-and-paper approach. It is true that much of university mathematics is driven by theorems and proofs, and it is also true that Python does not prove theorems. However, Python gives us the power and freedom to glimpse into the unknown, leading us towards insightful conjectures that would have been difficult to formulate otherwise. Hence, I was motivated to write a mathematics textbook that is richly interwoven with Python, rather than another Python textbook with some mathematical examples. The spirit of this book is one of mathematical exploration and investigation. I want to show students that Python can hugely enrich our understanding of mathematics through: • Calculation: Performing complex calculations and numerical simulations instantly; • Visualisation: Demonstrating key theorems with graphs, interactive plots and animations; • Extension: Using numerical findings as inspiration for making deeper, more general conjectures. Assumed knowledge: In terms of programming knowledge, this book does not assume that you are a highly experienced user of Python. On the other hand, complete beginners to Python might struggle to follow the code given. I would suggest that the reader should have the most basic knowledge of programming (e.g. you know what a for loop does). For completeness, I have included a section called Python 101 (Appendix A) which gives instructions on installing Python and signposts to references that can help anyone pick up Python quickly. In terms of mathematical knowledge, I do not assume any university-level mathematics. Students who are familiar with the material in the standard A-Level mathematics (or equivalent) should be able to follow the mathematical discussions in this book. This book is for all learners of mathematics, with the primary audience being mathematics undergraduates who are curious to see how Python can enhance their understanding of core university material. The topics chosen represent a mathematical overview of what students typically study in the first and second years at university, namely analysis, calculus, vector calculus and geometry, differential equations and dynamical systems, linear algebra, abstract algebra and number theory, probability and statistics. As such, it can also serve as a preview of university mathematics for high-school students. The prerequisites for reading the book are a familiarity with standard A-Level mathematics (or equivalent senior high-school curricula) and a willingness to learn programming. For mathematics lecturers and teachers, this book is a useful resource on how Python can be seamlessly incorporated into the mathematics syllabus, assuming only basic knowledge of programming.