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An Annotated History of the Reals

In the beginning, there was nothing, \( 0 = \emptyset \). Then we realized it was something, \( 1 = \{ \emptyset \} \). 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 ...


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 ...


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 ...


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 ...


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.


Area and Perimeter of Astroids

It is about time I revisited my first adventure Envelopes and Astroids. In the end of the post, I named the envelope of the line segments an Astroid. I will now find two basic properties of this shape: area and perimeter.


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