Fundamental Theorem of Arithmetic

Date: 07/08/97 at 18:44:42
From: bill
Subject: Fundamental Theorem of Arithmetic

What is the big deal?  I have excelled in math for many years, all the 
while ignorant of what I recently learned to be the "Fundamental 
Theorem of Arithmetic".

Date: 07/13/97 at 02:06:48
From: Doctor Chita
Subject: Re: Fundamental Theorem of Arithmetic

Dear Bill,

It's a philosophical thing, like enjoying a Mozart symphony without 
knowing anything about composition, or a poem without knowing anything 
about iambic pentameter, or art without knowing anything about 
Romanticism. Sure, you can combine numbers without quoting the 
fundamental theorem of arithmetic, solve equations without quoting the 
fundamental theorem of algebra, differentiate and integrate functions 
without quoting the fundamental theorem of calculus. But isn't 
learning all about the search for truth, beauty, and consistency?

The fundamental theorem of arithmetic is at the center of number 
theory, and simply, but elegantly, says that all composite numbers 
are products of smaller prime numbers, unique except for order. For 
example, 12 = 3*2*2, where 2 and 3 are prime numbers. The prime 
numbers, themselves, are unique, starting with 2. (1 is not considered 
a prime, since prime numbers have exactly two divisors; themselves 
and 1. The number 1 has only 1 divisor: itself.)

Like atoms in chemistry, prime and composite numbers are the building 
blocks of arithmetic. For a nice, non-technical discussion of number 
theory and its intriguing mysteries (for example, is there a largest 
prime number?), check out Ivars Peterson's book, _The Mathematical 
Tourist_. 

For me, the fundamental theorem of arithmetic is one of the few things 
I can depend on. I know I sleep better at night knowing it.

-Doctor Chita,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 07/13/97 at 12:25:40
From: Anonymous
Subject: Re: Fundamental Theorem of Arithmetic

Dr. Chita, 
  
Your answer to my inquiry about the fundamental theorem of arithmetic,
although elegant, just didn't do it for me. I understood what the 
theorem states, and perhaps now you have given me an appreciation for 
its simplicity, but I am looking for something more. What would you 
say to a student who asks, "Why did we just spend 20 minutes stating/
writing down/discussing this theorem? So what if every number can be 
expressed as a product of primes?"  

I would like to be able to explain/show the student where this theorem 
fits into the big picture: what it leads to, what important concepts
use it as a basis, why we would still be in the 18th century without 
it.
 
Thanks for your original response. I would also appreciate any further 
insight which might calm my frustrated engineer brain.

Bill Sanford

Date: 07/14/97 at 23:11:17
From: Doctor Chita
Subject: Re: Fundamental Theorem of Arithmetic

Dear Bill,

Now that you've centered the original question in a context, I can 
more easily appreciate your frustration. I know how students have a 
way of asking questions that can deflate many flights of fancy.

I am not sure that students understand the need that mathematicians 
have for defining their world as one that starts from a few basic 
premises. For example, in Euclidean plane geometry, there are just a 
few defined terms (e.g., point, line, plane, space) and a minimum 
number of postulates (axioms). For example, "A straight line can be 
drawn from any point to any point." and "All right angles are equal to 
one another." From this set of geometric building blocks, all else is 
derived in the form of theorems.

In the case of number theory, the subject is of course numbers - the 
natural numbers, that is: {1, 2, 3, . . . }. Number theorists want to 
know as much about them as possible. The specialness of the 
fundamental theorem lies in the fact that there there is an infinite 
set of prime numbers and that unique combinations of them generate an 
infinite set of composite numbers. The fact that there is no largest 
prime suggests that there is no largest composite number. Again, this 
is a sort of metaphysical observation. 

As for uses; modern cryptography relies on the fundamental theorem of 
arithmetic. By combining primes and forming very large composite 
numbers, code makers can create a key that makes a cipher virtually 
unbreakable. So, knowing how to construct numbers has a very practial 
use in our world today. If you will forgive another suggestion, you 
and your students might want to read a little book called _Cryptology_ 
by Albrecht Beutelspacher, published by the MAA. And of course, 
there's the book _Enigma_ about Turing (the father of artificial 
intelligence) and how he cracked the German's code during WW II.

Finally, arguments about the beauty of mathematics may fall on deaf 
ears when the audience consists of kids who have somehow picked up the 
idea that mathematics is all about solving arithmetic problems. With 
such a narrow view, they cannot yet appreciate that mathematics is a 
way of thinking, made up of many kinds of objects (numbers, equations, 
figures, etc.), and each conforming to a logical flow of postulates, 
definitions, and theorems.

Encourage your students to read more about math and mathematicians. 
Have them research Euclid's proof that there is no largest prime. What 
is a Mersenne prime? What is the current "largest" prime, discovered 
by a computer just a short while ago? And what is the product of this 
prime and a smaller prime? Maybe if you could get them to think more 
about the big ideas of numbers, they will gradually appreciate what 
makes me sleep more soundly at night.

Thanks for getting back. I don't know if I've helped you any further. 
I hope so. Good luck!

-Doctor Chita,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/