# An analysis of spacetime numbers

4-30-1999

7-19-2016

Thesis

## Degree Name

M.A. in Mathematics

Mathematics

## College

College of Science & Mathematics

Osler, Thomas J.

## Subject(s)

Space and time--Mathematical models

## Abstract

The familiar complex numbers begin by considering the solution to the equation i2 = -1, which is not a real number. A two-dimensional number system arises of the form z = x + iy. Spacetime numbers are based upon the simple relation j2 = 1, with the corresponding two-dimensional number system z = x + jt. It seems odd that anything useful can come from this simple idea, since the solutions of our j equation are the familiar real numbers +1 and -1. However, many interesting applications arise from these new numbers. The most useful aspect of spacetime numbers is in solving problems in the areas of special and general relativity. These areas deal with the notion of space-time, hence the name "spacetime numbers." My goal is to explain in a direct, yet simple manner, the use of these special numbers. I begin by comparing spacetime numbers to the more familiar complex numbers, then introduce the spacetime plane as a mathematical construction and show unusual features of spacetime arithmetic. A spacetime version of Euler's formula is then presented and then the solutions to the one-dimensional wave equation.

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