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  • Irrationally it’s Happy π Day!

    12–18 minutes

    Happy π Day! Today is a big day, it’s π\pi day! Which is basically THE math(s) holiday, also, just as significant, it’s Einstein‘s birthday! So in the spirit of π\pi day, we celebrate by doing some math(s). In particular, today, we will prove a few things that will culminate in a π\pi-related proof! First, as…

  • Euler’s Totient Function ϕ and Gauss’s Theorem

    9–13 minutes

    What I would like to show you today is one very interesting property that Euler’s totient function satisfies, which was discovered by Gauss, so the theorem is kind of an Euler/Gauss team-up! As it turns out, if you sum ϕ(d)\phi(d) over every d|n,d\mid n, then you get back n.n. That is, Where ∑d|n\sum_{d\mid n} means…

  • Playing With Pythagorean Triples

    18–27 minutes

    ***It’s recommended that you read this article on a computer or tablet. I apologize that the formatting isn’t great on a smartphone!*** Almost everyone learns the Pythagorean theorem in school. Today, we will discuss something you may or may not have learned about: Pythagorean triples! Just in case you haven’t, a Pythagorean triple is the…

  • Time Dilation Through Triangles

    12–17 minutes

    One of my friends made a small comment about how crucial geometry was to the development of math and science. It was that small comment that inspired this article. Furthermore, since everyone, including me, loves time dilation and special relativity, I thought this would be a great topic. Another reason I wrote this is that…

  • The AM-GM: Circles, Cauchy, and Pólya

    13–19 minutes

    ***It’s recommended that you read this article on a computer or tablet. I apologize that the formatting isn’t great on a smartphone!*** One of my favorite theorems (inequalities) is the AM-GM inequality (or simply the AM-GM). “Why?” I hear you ask through the screen, it’s because there are so many wonderful proofs of the darn…

  • Cosets are Cool Sets (Even I Know This is a Lame Title!)

    10–15 minutes

    *Note that it’s assumed the reader is familiar with groups and subgroups. * This article was initially written for a friend of mine when we were both taking abstract algebra 1 together. I hope you find it beneficial! Have you ever seen a movie where the ending was so amazing that you left the theater…

  • Thomae’s Function aka The Stars over Babylon aka The Popcorn Function

    7–11 minutes

    * Note it’s assumed the reader has a little familiarity with limits for sequences and functions, the connection between limits for sequences and functions (Called Heine’s Continuity Criterion), and continuity for functions. I haven’t decided whether to write a blog post about any of these topics. But it you want that, please comment so and…