A blind spot in undergraduate mathematics: The circular definition of the length of the circle, and how it can be turned into an enlightening example

TL;DR

The paper identifies a circular definition of pi in undergraduate calculus and proposes a method to resolve it, using this as an educational example to discuss mathematical foundations.

Contribution

It reveals a circularity in defining pi through circle length and offers a way to correct this, enhancing teaching approaches.

Findings

Highlights the circular definition issue in undergraduate math

Proposes a method to break the circularity in pi's definition

Uses the problem as an educational tool for teaching mathematical foundations

Abstract

We highlight the fact that in undergraduate calculus, the number pi is defined via the length of the circle, the length of the circle is defined as a certain value of an inverse trigonometric function, and this value is defined via pi, thus forming a circular definition. We present a way in which this error can be rectified. We explain that this error is instructive and can be used as an enlightening topic for discussing different approaches to mathematics with undergraduate students.