Banach-Tarski Paradox

It is a common occurrence in mathematics that when something does go wrong, it goes terribly wrong. This exact phenomenon occurs with the Banach-Tarski Paradox. Informally, it says that one can take a sphere (in 3 or more dimensional space) can be split into finitely many pieces and, using only rigid motions, can be rearranged […]

Pythagoras and the Square Root of 2

Pythagoras of Samos (569-500 BCE) was an actual person, but was also the founder of the Pythagoreans. He was a political figure and a mystic. Beyond this, he stood out in his time as he involved women as equals in his activities. The Pythagorean society focused on mathematics, but also had some religious properties which include […]

The Volume of a Sphere with Calculus

This post is second in a series of 3 beginning with “Coordinates in 3-space”. We have been taught that the volume of a sphere . We will go through two derivations of this, beginning with the Calculus version. We will make use of a theorem in calculus. Theorem: . Suppose we have a sphere of radius […]

Paradoxes: The Birthday Problem

As living beings we rely heavily day to day on our intuition. And most of the time it does not let us down. However, sometimes our intuition fails us and we feel compelled to call that event a paradox. This is the case with the birthday problem. I pose the following question. How many random […]

The Road to Larger Infinities

Georg Cantor was a German mathematician responsible for the invention of set theory. Before him, the concept of infinity was not well studied or considered by mathematicians. The importance of Cantor’s development of set theory is very clear as it is the basis for much of modern mathematics. We will take a look at Cantor’s […]

Infinitude of Primes

It is well known that there are infinitely many primes. The proof by Euclid is among the first we see as rising mathematicians. Though it is often put into a proof by contradiction context, the original proof does not actually contain a reducto ad absurdum argument. Below we will share Euclid’s famous proof and a […]