Capstone Studies for Math Majors via Complex and Hyperbolic Numbers

Abstract

This article is a discovery-based instructional resource for faculty to use as a capstone course or exploratory project for undergraduates who are familiar with (but not necessarily fluent in) calculus, linear algebra, complex variables, and geometry. The explorations flow from a simple change in sign: in the complex numbers a + bi, where $i^2=-1$, replace i by τ, where ${\tau }^2=1$, but where $\tau \ne \pm 1$. This creates a new algebra and geometry that is counter-intuitive, but has parallels to familiar mathematics so that students can use what they know, but on different ground. The materials are intended to foster investigation, collaboration, and creative mathematical thinking.

Keywords:

• Capstone course
• complex numbers
• geometry
• hyperbolic numbers
• Lorentz numbers
• split-complex numbers
• rotation

ACKNOWLEDGMENTS

The authors appreciate the thoughtful comments of the referees, which helped to improve the paper.

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the authors.

Additional information

Notes on contributors

Rachid Atmai

Rachid Atmai has studied and taught mathematics internationally, including graduate work in France and a faculty appointment in Vienna. He likes to look at teaching from the bigger picture of sparking students' curiosity and interest through asking good questions and looking for answers from a research-oriented mindset.

Curtis Kizer-Pugh

Curtis Kizer-Pugh likes to explore new facets of math and find ways to relate them. He is a recent graduate of the University of Texas at Arlington in physics and mathematics and a McNair scholar. He is continuing with graduate studies in mathematics at Louisiana State University. In the words Herman Melville, “I try all things; I achieve what I can.”

Barbara Shipman enjoys helping people discover beautiful ideas in mathematics in fun and creative ways. She especially enjoys teaching seminar courses for undergraduates where students find, explore, and share their own questions and ideas. She is a Distinguished Teaching Professor at the University of Texas at Arlington.