Proceedings of Symposia in Applied Mathematics

Volume 46, 1992

The Unreasonable Effectiveness of Number Theory

in Physics, Communication and Music

Manfred R. Schroeder

ABSTRACT. This paper samples applications of number

theory in the real world, ranging from concert hall acoustics

to general relativity.

1. Introduction

Number theory has been considered since time immemorial to be the very

paradigm of "pure" (some would say useless) mathematics. According to

Carl Friedrich Gauss, "mathematics is the queen of sciences - and number

theory is the queen of mathematics.'' What could be more beautiful than a

deep, satisfying relation between whole numbers. (One is almost tempted to

call them wholesome numbers.) Indeed, it is hard to come up with a more

appropriate designation than their learned name: the integers - meaning the

"untouched ones." How high they rank, in the realms of pure thought and

aesthetics, above their lesser brethren: the real and complex numbers - whose

first names virtually exude unsavory involvement with the complex realities

of everyday life!

Yet the theory of integers can provide totally unexpected answers to real-

world problems. In fact, discrete mathematics is taking on an ever more

important role. If nothing else, the advent of the digital computer and digital

communication has seen to that. But even earlier, in physics, the emergence

of quantum mechanics and discrete elementary particles put a premium on the

methods and, indeed, the spirit of discrete mathematics.

1991 Mathematics Subject Classification. Primary 11A07, 11B83.

This paper is in final form and no version of it will be submitted for publication elsewhere.

© 1992 American Mathematical Society

http://dx.doi.org/10.1090/psapm/046/1195839 http://dx.doi.org/10.1090/psapm/046/1195839