All Numbers Great and Small?
In physics, different laws apply on the tiny scale of atoms than apply on the large scale of people and planets. We'll answer a listener's question about whether scale matters in mathematics -- after this on Earth and Sky.
Date
DB: This is Earth and Sky, with a listener's question. Ameera Chowdhury writes, "I know there are different laws of physics for the small scale of atoms and the large scale of planets. Is the same true for math -- do the same rules apply for small numbers as for large?" Ameera, we asked Dr. Hendrik Lenstra, a mathematician at the University of California at Berkeley:
(Tape 0:07:18-0:08:18) If you look at the small numbers and you sort of picture them as dots on a line and you have one, and then a little further you have two and you have three, and they are sort of individuals and each of them has its own properties. But if you look at large numbers, well then you somehow have to picture them in the distance and then they all dissolve into a gray mass and you find it much harder to tell them apart. And it has certainly happened many times in number theory that people were curious as to a certain theorem being valid for all numbers. And what they would do is just try it out by computations, nowadays on a computer, and they would find that certain things would always be true for all numbers up to a thousand, up to a million, up to a billion, and then they would proudly formulate the conjecture, well the thing is always true. And then there would be some theoretical guy who would show that the thing is just wrong as soon as your number is going to have four hundred digits. (Tape 0:08:23-0:08:29) So I think the answer is yes, it is just like in physics you have small scale phenomena and large scale phenomena.
DB: Thanks to Dr. Hendrik Lenstra for speaking with us. And with thanks to the National Science Foundation, I'm Deborah Byrd, for Joel Block, for Earth and Sky.
Author: Beverly Wachtel
Thanks to the following individual who aided in the preparation of this
script:
Dr. Hendrik W. Lenstra
Department of Mathematics
University of California
Berkeley, CA
hwl@math.berkeley.edu
If you enjoyed this program, you may be interested in the following websites:
Mathematical Sciences Research Institute:
http://www.msri.org
Introduction to Number Theory:
http://www.math.niu.edu/~rusin/known-math/index/11-XX.html
Math Archives - Number Theory:
http://archives.math.utk.edu/topics/numberTheory.html
Number Theory Web (American site):
http://www.math.uga.edu/~ntheory/web.html