## Period Three Implies Chaos

Using Intermediate Value Theorem to prove that period-3 implies all other periods. Also, Sharkovsky's theorem is stated.

## An Annotated History of the Reals

In the beginning, there was nothing, \( 0 = \emptyset \). Then we realized *it* was some*thing*, \( 1 = \{ \emptyset \} \). Then we wondered, why not have another thing? \(2\). And another thing, \(3\). And another, \(4\), and another, \(5\), and another, \(6\), etc. Just like that, we had the **natural numbers**!^{1} And …

## 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 - a beautiful identity.

## 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 was not …

## 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 memoir …

## 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 …

## Inclusion-Exclusion Principle (Pt. 1)

The Inclusion-Exclusion principle is one of those things I understood quickly and intuitively; in fact, it seemed obvious to me. It was not until I tried to prove it that I was amazed by its dependence on the binomial theorem and also realized it was not so obvious after all …

## Schinzel's Theorem

Today, I'll translate the proof for a theorem by André Schinzel that gives a circle with $n$ integer coordinates (also known as **Lattice Points**) on its circumference, for any given $n$. The theorem was published in *L'Enseignement Mathématique* in 1958 and can be found here.