A second look at probability theory

I'm studying Principles of Communication Engineering by Wozencraft and Jacobs and the second chapter goes over probability theory, except it uses measure theoretic probability theory.
Here are my notes when reading about σ-algebras (sigma-algebra).

RFC 6090: Fundamental Elliptic Curve Cryptography Algorithms

RFC 6090: Fundamental Elliptic Curve Cryptography Algorithms

This note describes the fundamental algorithms of Elliptic Curve Cryptography (ECC) as they were defined in some seminal references from 1994 and earlier.  These descriptions may be useful for implementing the fundamental algorithms without using any of the specialized methods that were developed in following years.  Only elliptic curves defined over fields of characteristic greater than three are in scope; these curves are those used in Suite B.

Five proofs that the squrate root of 2 is irrational

This weekend I went through a few proofs that the square root of 2 is irrational.

1. The elementary proof is the one that everyone sees. You assume
   that \(\sqrt{2}\) can be represented as \(\frac{p}{q}\) where \(p\)
   and \(q\) share no common factors, that is \(\gcd(p,q)=1\). By
   multiplying out both sides you're lead to the result that both
   \(p\) and \(q\) must have 2 as a factor. See
   Wikipedia for the proof.
2. The second proof is a stronger result, and relies on the
   fundamental theorem of arithmetic to prove that \(\sqrt{p}\)
   where \(p\) is a prime number must be irrational.
3. The third proof again relies on the fundamental theorem of
   arithmetic and states that \(\sqrt[k]{n}\), the k-th root of a
   number \(n\) that is not a perfect k-th power is irrational.
4. The fourth proof is one that I hadn't seen before. It is
   elementary and of course Wikipedia has it. I like the one I found
   in Galois Theory by Ian Stewart:
   1. Assume that there exist integers a, b with \(b \neq 0\) such
      that \((a/b)^2=2\).
   2. Show that we may assume \(a,b > 0\).
   3. Observe that if such an expression exists, then there must be
      one in which \(b\) is as small as possible.
   4. Show that \((\frac{2b-a}{a-b})^2 = 2\).
   5. Show that \(2b-a>0,a-b>0\).
   6. Show that \(a-b<b\), a contradiction.
5. The fifth proof uses the rational root theorem on the
   polynomial \(x^2-2=0\).