# The Transitivity of Whole Numbers

Introduction

The less than relation is an order relation of real numbers.  In this post, however, to prove one of its properties, we limit it to the set of whole numbers.  In the set of whole numbers, $latex a$ is less than $latex b$, or $latex a < b$ in symbol, if and only if there exists a whole number $latex m$ such that $latex a + m = b$. For example, we are sure that $latex 9 < 11$ because there exists a whole number $latex 2$ such that $latex 9 + 2 = 11$.

Below we prove one of the properties of order relations in the set of whole numbers.  That is, if $latex a < b$ and $latex b < c$, then $latex b < c$. This is called the transitivity property for the relation $latex <$. Continue reading

# Proof that Square Root of Three is Irrational

Introduction

In The Intuitive Proof of the Infinitude of Primes, I showed you a proof strategy called proof by contradiction.  In this post, we use this strategy to prove that $latex \sqrt{3}$ is irrational.  In proof by contradiction, if the statement P is true, you have to assume the contrary, and then find a contradiction somewhere. Note that the proof in this post is very similar to the proof that $latex \sqrt{2}$ is irrational.

Theorem:  $latex \sqrt{3}$ is irrational. Continue reading