Analogies are the Worst!
Why you should be careful when giving analogies/examples in a debate. I use meta-analogies and it's totally not ironic.
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 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 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 …