You are currently browsing the monthly archive for November 2011.

“You don’t have to believe in God, but you have
to believe in The Book.”—Paul Erdős

Paul Erdős liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics.

JOEL SPENCER: “Paul talks about The Book. The Book has all the theorems of mathematics. Theorems can be proven in a lot of different ways, but in The Book there is only one proof and it is the one that is the clearest proof, the one that gives the most insight, the most aesthetic proof. It’s what he calls The Book proof. And sometimes when there’s a problem and somebody solves it and the proof is not so beautiful, then he’ll say, “Well okay, but let’s look for The Book proof; let’s try to find The Book proof.” And this is the sense of mathematics, that…that The Book is there, the theorems have an existence of their own. And what we’re doing is we’re just trying to uncover. We’re trying to read the pages of The Book. We don’t create mathematics. What we do is we read the pages of The Book. We discover the pages of The Book. So when he goes from university to university, and he talks about problems, and he asks everybody to try to solve these problems, it doesn’t matter who solves the problem. It really doesn’t matter to him, because all of us are in the same venture. We’re all just trying to uncover the pages. And sometimes we succeed. Sometimes we find these beautiful theorems.”
View the video of Joel Spencer describing The Book here.

Proofs from THE BOOK is an effort by Martin Aigner and Günter Ziegler to reveal an approximation to a portion of The Book

Paul Erdős Biography [1, 2, 3]

“My brain is open” – Paul Erdős

Paul Erdős was born on March 26, 1913 in Budapest, Hungary to a Jewish family whose name was originally Engländer. Though the times of antisemitism was behind them, the Hapsburgs did not want to be reminded of their Jewish neighbors. Thus, Paul’s father picked a common Hungarian name that means “from the woods”. It’s approximate pronunciation is air-dish. His parents Lajos and Anna had two daughters who died just days before Paul was born. This would make his mother extremely protective of Paul. He would get his introduction to mathematics from his parents who were both mathematics teachers; a profession that was held in high regard in Hungary which boasted an outstanding educational system.

It turned out that Paul was a childhood prodigy who had an affinity for numbers. He would learn to count when his mother left for teaching. One day when Paul was just four years old a visitor, who after Paul had calculated the number of seconds he had lived, decided to give Paul a tricky question. He asked, “What is 150 minus 200?” Paul went quiet for a moment as his mind went of into unknown territory. Then he smiled and yelled excitedly, “150 below zero!” This was no small feat. He just independently discovered negative numbers! He would later down play his calculating abilities, but he would always remark with pride “his discovery” at the age of four.
Read the rest of this entry »

Proposition 1 Prove ${\sqrt{2}}$ is irrational.

Here is a proof using a traditional method (See Euclid’s Elements Book X which incorporates Theatetus work on incommensurable numbers. It includes a proof that ${\sqrt{2}}$ is irrational (Proposition 22), and ends with a proof that there are infinitely many distinct irrational numbers (Proposition 115): Read the rest of this entry »

Riker and Picard discuss Fermat’s Last Theorem in “The Royale.” Captain Picard: ‎”For 800 years people have been trying to solve it (Fermat’s Last Theorem)…”

Andrew Wiles did solve it after more than 350 years after Fermat first mentioned the proposition in the margin of his Arithmetica in 1637. Wiles proved FLT in 1994 and it was published in 1995. Apparently, the writers of ST TNG thought this theorem was completely inaccessible.

Hei-Chi Chan’s first book:

World Scientific, May 25, 2011 – Mathematics – 236 pages
The aim of this lecture notes is to provide a self-contained exposition of several fascinating formulas discovered by Srinivasa Ramanujan. Two central results in these notes are: (1) the evaluation of the Rogers–Ramanujan continued fraction — a result that convinced G H Hardy that Ramanujan was a “mathematician of the highest class”, and (2) what G H Hardy called Ramanujan’s “Most Beautiful Identity”. This book covers a range of related results, such as several proofs of the famous Rogers–Ramanujan identities and a detailed account of Ramanujan’s congruences. It also covers a range of techniques in q-series.

Preview the book here.

In college, we often learn of many infinite series that give the value of pi including one called the Leibniz series, named after Gottfried Leibniz. It is also called the Gregory–Leibniz series, recognizing the work of James Gregory. This unnecessarily attributes the discovery to the west, however, the formula was first discovered in India by Madhava of Sangamagrama and so is also called the Madhava–Leibniz series. Indian mathematicians made vast and fundamental contributions to our modern mathematics.

$\pi=4\sum_{k=0}^{\infty}\frac{(-1)^{k}}{2k+1}$

Knoppix/MATH is a LIVE linux distribution that contains many mathematical software. You can easily enjoy many mathematical applications and you don’t need to install any software. Knoppix is based off Debian Linux.

IMAGE: This knot has Gauss code O1U2O3U1O2U3.

In his article “The Combinatorial Revolution in Knot Theory”, to appear in the December 2011 issue of the Notices of the AMS, Sam Nelson describes a novel approach to knot theory that has gained currency in the past several years and the mysterious new knot-like objects discovered in the process.

As Nelson reports in his article, mathematicians have devised various ways to represent the information contained in knot diagrams. One example is the Gauss code, which is a sequence of letters and numbers wherein each crossing in the knot is assigned a number and the letter O or U, depending on whether the crossing goes over or under. The Gauss code for a simple knot might look like this: O1U2O3U1O2U3.

In the mid-1990s, mathematicians discovered something strange. There are Gauss codes for which it is impossible to draw planar knot diagrams but which nevertheless behave like knots in certain ways. In particular, those codes, which Nelson calls *nonplanar Gauss codes*, work perfectly well in certain formulas that are used to investigate properties of knots. Nelson writes: “A planar Gauss code always describes a [knot] in three-space; what kind of thing could a nonplanar Gauss code be describing?” As it turns out, there are “virtual knots” that have legitimate Gauss codes but do not correspond to knots in three-dimensional space. These virtual knots can be investigated by applying combinatorial techniques to knot diagrams. Read the article here.

In mathematics, a knot is defined as a closed, non-self-intersecting curve that is embedded in three dimensions and cannot be untangled to produce a simple loop (i.e., the unknot). While in common usage, knots can be tied in string and rope such that one or more strands are left open on either side of the knot, the mathematical theory of knots terms an object of this type a “braid” rather than a knot. To a mathematician, an object is a knot only if its free ends are attached in some way so that the resulting structure consists of a single looped strand.

However, give me any closed loop knot and an extra dimension and I will show you an unknot (i.e., there are no knotted 1-dimensional spheres (strings) in dimension 4. Zeeman took this further and stated that any n-dimensional sphere is an unknot in a space of dimension higher than $(\frac{3}{2})(n+1)$). Read the proof here.

UCLA mathematicians devise an algorithm based on data from the Los Angeles Police Department for the Hollenbeck area east of downtown. Read the article here.

This post was created using LaTeX to WordPress with the Terry Tao style. LaTeX2WP is a program that converts a LaTeX file into something that is ready to be cut and pasted into WordPress. It was compiled with Python Software Foundation’s Python 2.7.2 64 bit version.

This way, you can write, and preview, your post in LaTeX, then run LaTeX2WP, and post into WordPress whatever comes out.

Here is the post that was copied and pasted using the example.tex supplied by downloading the program:

Look at the document source to see how to strike out text, how to use different colors, and how to link to URLs with snapshot preview and how to link to URLs without snapshot preview.

There is a command which is ignored by pdflatex and which defines where to cut the post in the version displayed on the main page Read the rest of this entry »

This is a typical proof one may see in a undergraduate introduction to proofs mathematics course.

Proposition 1 Let the constant ${R}$ be defined such that

$\displaystyle R:=\frac{1}{F_{1}}+\frac{1}{F_{2}\cdot F_{1}}+\frac{1}{F_{3}\cdot F_{2}\cdot F_{1}}+\frac{1}{F_{4}\cdot F_{3}\cdot F_{2}\cdot F_{1}}\cdots$

and ${F_{n}}$ are the Fibonacci numbers where ${F_{n+k}>F_{k}}$ for ${n>0}$. Then ${R}$ is irrational.

Harvard Grad Glen Whitney starts a Math Museum, a 19,000 square foot space on East 26th Street in Manhattan and plans to open the doors in 2012. Whitney has raised about \$22 million from some 300 donors such as Google Inc., the Alfred P. Sloan Foundation and a charity founded by Simons. Whitney hopes to eventually create a museum national in scope with a broad donor base. Read about it here.