## Tuesday, January 31, 2012

### 11 Bitcoins = 1 \$ !!

By pure coincidence as I was running through the list of stack exchange sites I found a site called Bitcoin. I didn't understand what was it at first but when I looked it up in wikipedia it turned out to be an e-cash P2P system. Wikileaks, freenet, and another company already use it for reception of donations and for investments according to wikipedia. It even has an exchange rate to US dollars. The original paper is http://bitcoin.org/bitcoin.pdf and there are other more recent papers.
#WOW

## Friday, January 20, 2012

### Proof that np=mq and mn>0 implies pq>0

Implicit assumptions: q != 0 (p/q is a rational number).

This is a proof that mn>0 as a definition that a rational number m/n is "positive" is a well-defined with respect to equivalence classes of rational numbers. I.e. if p/q is in the same equivalence class of m/n (i.e. np=mq) and m/n is positive (mn>0) then p/q is also positive (pq>0).

First, np=mq => np-mq=0, but since mn>0 (neither m nor n is zero), and that q!=0 (by definition), then p!=0. So one of pq<0 or pq>0 is true. Assume the former towards a contradiction. First let p>0 and q<0 and let q=-q' where q' is positive. Then 0=np-mq=np+mq', but all of these are positive, so they cannot equal zero: contradiction. Same argument for p<0 and q>0, we end up with 0=-np'-mq, but the right hand side is negative: contradiction. So pq<0 cannot be true, therefore pq>0.