Wednesday, April 12, 2006

Ask and ye shall receive

In response to this post Harvey asked, "what's the Fundamental Theorem of Arithmetic?" To some extent I think that the name is a bit of a misnomer, but for now I'll give you the theorem without proof. The proof is relatively simple and is interesting in as much as it illustrates the process of mathematical thinking. But before I entertain you with that, I need to sit down and do a little bit of work to provide comprehensibility for the non-math people.

The Fundamental Theorem of Arithmetic Every positive integer greater than 1 can be written uniquely as a product of primes, with the prime factors in the product written in nondecreasing order.

For example, the prime factorization of the positive integer 1001 = 7 * 11 * 13.



5 comments:

Anonymous said...

I hear her keyboard moving but all I see is clickity click click clickity *eyes glazed over* maybe I just need coffee. Nah.

Teresa said...

Considering it's a product... does the order matter?

MathCogIdiocy said...

Technically, order doesn't matter since multiplication, at least in this case, is commutative. I think that the theorem's statement - the prime factors in the product written in nondecreasing order - is more indicative of an underlying thought process. This will make more sense when I get the proof up and concepts like "well ordered" show up.

Harvey said...

*blink*

Somehow I was expecting something more earth-shattering. To my non-mathematician's ear, this sounds a lot like "water is wet" :-)

Anyway, I'm curious to see why this theorem is held in such high esteem.

By the way, is it possible to construct a consistent form of mathematics using the opposite theorem (NO positive integer greater than 1...) - sorta like non-Euclidian geometry?

MathCogIdiocy said...

Earth-shattering in math is so abstract that ... nevermind. :-) Much of the early number theory is a lot like this theorem - statements of apparently obvious things with rather involved proofs.

I have to think about this -
"By the way, is it possible to construct a consistent form of mathematics using the opposite theorem (NO positive integer greater than 1...) - sorta like non-Euclidian geometry?" Unfortunately as far as math is concerned, constructing a system and showing that it's consistent is rather complicated. Easier to show inconsistency.