Misadventure #1

Now seems like a good time to write about my failed math research. There are several questions I've never been able to answer and today I present the most frustrating one - an identity. $$\left(1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{9…

Hmm, Ordinal Numbers...

In the last post, Why Ordinal Numbers, I had intuitively guessed that I would be left with a perfect set after enough application of derived sets on ${S}$. By the end, I gave an example of a set that had a nonempty “infinite” derived set, ${S^{(\infty)},}$ that…

Why Ordinal Numbers?

It's about time I write on a non-Euler topic -- the birth of ordinals. I've been reading Georg Cantor's Contributions to the Founding of the Theory of Transfinite Numbers. In two papers published in 1895 and 1897, Cantor creates the theory of cardinals and ordinals from the ground up. The…

Basel's Problem - New proof and some calculus

Sorry for the long hold up on another post. This new post is a result on my investigations on a proof of ${\zeta(2) = \frac{\pi^2}{6}}$ by Daniele Ritelli. Yes, it's another post on something Euler solved first. But I did things different this time around: I discussed…

From Heron's formula to Descartes' circle theorem

Descartes' Circle Theorem is a very remarkable and simple statement and, to prove it, I had to use three other surprising theorems of the triangle, including Heron's formula. While I had been aware of Heron's formula before, it was during my research on Descartes' theorem that I discovered the inradius…

Inclusion-Exclusion Principle (Pt. 2)

Last post was a proof for the Inclusion-Exclusion Principle and now this post is a couple of examples using it. The first example will revisit derangements (first mentioned in Power of Generating Functions); the second is the formula for Euler's phi function. Yes, many posts will end up mentioning Euler…