Geometry MMA Seminar: The ups and downs of negativo-affirmative arithmetick

Speaker:  Joseph Vandehey, University of Texas at Tyler

Title:  The ups and downs of negativo-affirmative arithmetick

Abstract:  In the early 1700s, John Colson proposed to perform arithmetic by allowing digits (not just numbers) to be either positive or negative. So instead of writing $187$, we could write $(2)(-1)(-3)$. This idea has impacts in the modern day, with applications to cryptography and computing. We will discuss some of the strange dynamical properties of these expansions, and how we deal with understanding the structure of space when the representations of points are no longer fixed. Included will be a discussion of recent work of the speaker, examining the relation between hyper-$n$-ary expansions (which allow digits $\{0,1,\dots, n\}$) and balanced $n$-ary expansions (which allow digits $\{-n/2, -n/2+1, \dots, n/2\}$ for even $n$). This talk will contain a surprise guest appearance by continued fractions.

Time: Monday, October 21, 2024 – 10:30am-11:30am

Place: Exploratory Hall, Room 4106