Student-Faculty Colloquium Some Proofs of the Existence of Irrational Numbers

Over the course of this article, we will discuss irrational numbers and several different ways to prove their existence. As is commonly known, the real numbers can be partitioned into rational numbers and irrational numbers. Rational numbers are those which can be represented as a ratio of two integers — i.e., the set { b : a, b ∈ Z, b 6= 0} — and the irrational numbers are those which cannot be written as the quotient of two integers. We will, in essence, show that the set of irrational numbers is not empty. In particular, we will show √ 2, e, π, and π are all irrational.