i love this stuff!

patric sent this to me via surfapig@live.com: Hey Mike,

Its Patrick (Capitola Dewey Weber Pig-September 5, 2009 post). I wanted to share some photos my father shot of me surfing my pig last weekend. I like reading your blog and wanted to make another contribution. Keep mining the stoke, Patrick Cady P.S. I have a math question for you. Is there a way to prove that every rational number can be expressed as a fraction of two coprime integers. I have been taking this for granted as part of the definition of rational numbers, but from what I can tell the coprime part isn't in the definition. I'm writing a proof for root 2 is irrational (by contradiction).

needless to say, i love the photo contributions, and was stoked to get the math question!!!! here was my response:

The fact that your ratio in the "square root two" proof is a reduced fraction (a ratio of coprime integers) is essential. It is the fact that is contradicted.

More generally speaking, we build the rational numbers from the integers. integer over non zero integer . it is a theorem that every integer has a unique factorization. So, since I can factor every integer one and only one way (as a product of primes) , and since any number over itself is the value "one". and since one is the identity element of multiplication, it follows that any rational number can be written as a ratio of co prime integers.