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.
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