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 is a rather good read since it assume no prior knowledge of set-theory on part of the reader; in fact, set theory had not even been properly formed at this time. Unfortunately, since it is a foundational text (if that makes any sense), you get the feeling he's forming all this great ideas out of thin air. This post documents my journey to convince myself how ideas of ordinal numbers can arise naturally when studying point-sets, specifically, sets of real numbers. Continue reading

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 the few (advanced) Calculus theorems that played crucial roles in the proof.

Theorem. \displaystyle \qquad \zeta(2) = \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}

Continue reading

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 and exradius formulas. The circle theorem was first described by Descartes in 1643 and then rediscovered by Philip Beecroft in 1842, Frederick Soddy in 1936, and M. E. Wise in 1960. To be exact, Wise actually discovered the three dimension version of the theorem unaware of the plane version. Descartes' circle theorem has a history that was as interesting to read as its various proofs. Without further ado... Continue reading

The 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. In this post, I chose to take an unorthodox path and present the theorem at the end of the post rather than at the start. Continue reading

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. Continue reading

The Power of Generating Functions

In Pentagonal Number Theory, I touched on the topic of generating function but now I'll give examples of generating functions being used to find explicit solutions for recurrent relations. I think this was their primary purpose; Abraham de Moivre invented them when he tried to find the exact formula for the {n^{th}} term of a sequence defined by a linear recurrence relation, like the Fibonacci Sequence.

Definition 1. The Fibonacci Sequence is {1,1,2,3,5,8,13,\ldots} defined by the following recurrence:

  F(0)=1; \quad F(1)=1; \quad F(n)=F(n-1)+F(n-2), ~ n\ge2

Binet's Formula.

  F(n) =\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}\right)^{n+1}-\frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^{n+1}

Sidenote: It is named after Binet even though de Moivre discovered the formula a century before him. Continue reading

Surprising Surds

Yesterday, I stumbled upon this very surprising identity while reading on nested radicals:

  \sqrt[3]{2\pm\sqrt{5}} = \frac{1\pm\sqrt{5}}{2}

While it is easy to prove the fact after seeing it, I will try to prove this from the perspective of someone who is not sure if {\sqrt[3]{2\pm\sqrt{5}}} is the simplest form of the pair. Continue reading

Euler's Pentagonal Number Theorem

Today, I'll prove Euler's Pentagonal Number Theorem and show how he used it to find recurrence formulae for the sum of {n} 's positive divisors and the partitions of {n} . This post will be based on two papers I read last week: “An Observation on the Sums of Divisors” and “Euler and the Pentagonal Number Theorem”.

Definition 1. The generalized pentagonal numbers are those of the form

  p(n) = \dfrac{3n^2-n}{2}

for any integer {n} .

Pentagonal Number Theorem.

 \begin{aligned}(1-x)(1-x^2)(1-x^3)\cdots &= 1 - x - x^2 + x^5 +x^7- x^{12} -x^{15}+x^{22}+\cdots \\ &\text{or} \\ \prod_{i=1}^\infty (1-x^i) &= \sum_{i \in {\mathbb Z}} (-1)^i x^{p(i)} \end{aligned}

Apparently, Euler conjectured this theorem in {1740} (or earlier) but it was not until {1750} that he proved it; Continue reading