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### Fundamental Theorem of Arithmetic

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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".
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```
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/
```
Associated Topics:
High School Number Theory

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