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Tuesday, February 28, 2012

Secondary school mathematics books

#Publishers #Freedom #Internet

How many secondary school mathematics books have been written since, say 1900 ? I don't know but I am sure there are many. I wonder what the authors of these books had in mind when they wrote these books. Were it the results of students? Orders of their (publisher-)employers? Or plain greed? Mathematics does not change because time passes. And what was the role of the department of education? How are they linked to the publishing industry in your country?

I am sure that everyone agrees that Gauss and Riemann received an excellent secondary school education, and neither were born in a rich family, to the contrary in fact. They had genius minds of course, but mathematics was flourishing in these days. What books did they study, I wonder? I am not sure, but for geometry they must have studied the works of Euclid, and for algebra and number theory they likely studied works of Euler.

Almost everything high school students have to learn for algebra is in Euler's books, really. All these books need is a modern e-book look and feel and you are done. And Euler starts really at the beginning, it's all there. A 10 year old can start studying maths from one of Euler's books. We don't need more books of the same. Do we? Have I mentioned that Euler's books are free? Does that ring a bell?

Euler's Elements of Algebra.


Monday, February 27, 2012

Publishers and the trebling of Open University fees


In the UK fees for Open University courses trebled. In the USA students take mortgage size loans to get a basic college education. The Open University terminated the employment of all continental European staff. ( ... ) The list is endless. - In the mean time there is one party in the field of education that keeps raking in profits, the publishers. Publishers are about as evil as bankers.

In this first post about the evil publishers I want to share some sounds with you that I picked up from the BlogoSphere.

The so-called publishing houses are the mafia. They're parasites on society and care solely about profit. They don't care one bit about human betterment so long as they can live in mansions and not have to work. It's well established that academics don't make any money for publishing, yet we're the ones doing all the work! And then, we're forced to pay for it (again) through massive licensing fees paid to publishing houses in exchange for having access to journals that our colleagues are forced to publish in. The entire system is rigged.

A lot of people really believe that was about stealing books and defrauding authors. No. It was about making vitally important information available to promote human flourishing. It is in everyone's best interest to have a highly educated population. As for the ridiculous argument that creators of creative works were being robbed, I've got to ask: Where did the ideas for a mathematical theorem come from--an individual genius working in isolation, bereft of many years of training by academic mentors? I think not. How do songs get created? Is it from the imagination of a single artist? Or does it entail a filtering of massive quantities of information from experiences in the environment through an individual brain? If it's the latter, why should we give so much credit to an individual where we should all take credit? We don't achieve in isolation. We're in this life together, and it seems extraordinarily unfair and cruel to me that a small group of individuals should profit at everyone else's expense under the mythology of "capitalism" (a facade for economic fascism) and "copyright."

The very idea of copyright is a scam from the get-go. It's merely a thinly veiled ploy to prevent others from gaining access to a work. And why would one want to do that? Status, power, and profit. "I'm better than you." "I deserve more than you." "My Porsche in Los Angeles is more important than you, over there in Africa, dying of hunger." It's really quite an arrogant and destructive idea. If I had a MacBook, and you took it from me without my permission, that would be stealing. But if you copy an ebook, you're merely replicating digital bits. It essentially costs nothing, and it doesn't deprive anyone of anything. To the contrary, it's a benefit because it increases the probability that someone along the line will read that ebook, learn something from it, and contribute to society. wasn't a cunning method of making money. It was a morally heroic effort to make the world better, and it was succeeding until the Nazis attacked it from a German court.

If a U.S. company developed a drug that victims of AIDS could take for one year and be cured, at a cost of 100,000 USD, yet millions of Africans were dying of AIDS, would it be morally right for the Africans to break the U.S. company's patent on the drug, manufacture it, and distribute it to everyone who needed it? I claim that that would be the morally right thing to do. Shockingly, a large number of people disagrees!

Dostoyevsky is dead, yet publishing houses are making millions of dollars on his books. Something has gone terribly wrong in our society. Contrary to the rhetoric of the media companies who sway public opinion by throwing money at it, pirates are needed to prevent the fascism and parasitism that has historically been the norm for the aggressive Homo sapiens sapiens species. Human progress, and what we call culture, are precarious and can disappear very quickly if the circumstances are right. Against the threat of descent into barbarism is education, to which contributed immensely on a global scale.

We must fight for freedom and progress.

Wasn't it Homer who somewhere wrote, "...for he was excellent above all men in theft and perjury?"

What an apt description of the plutarchs who run the publishing, music, film, and every other industry, and prey upon the rest of us who actually work and create.

Phil ( address of publication known by me )

From a post of Gowers's Weblog

The Dutch publisher Elsevier publishes many of the world’s best known mathematics journals, including Advances in Mathematics, Comptes Rendus, Discrete Mathematics, The European Journal of Combinatorics, Historia Mathematica, Journal of Algebra, Journal of Approximation Theory, Journal of Combinatorics Series A, Journal of Functional Analysis, Journal of Geometry and Physics, Journal of Mathematical Analysis and Applications, Journal of Number Theory, Topology, and Topology and its Applications. For many years, it has also been heavily criticized for its business practices. Let me briefly summarize these criticisms.

1. It charges very high prices — so far above the average that it seems quite extraordinary that they can get away with it.

... More at Gowers's weblog

To be continued.

Friday, February 24, 2012

Two mystery mathematicians.


Facts about two mathematicians.

His collected works appear in five volumes: the first contains 62 papers which (...) ; the second contains 107 of the 147 papers (...); the third includes 89 of the 180 papers (...); the fourth contains 98 of the 232 papers he published (...).

(...) school reports began to describe him as singular, bizarre, original and closed. - (...) took the examination of the École Polytechnique but failed. - This is the only student who has answered me poorly, he knows absolutely nothing. I was told that this student has an extraordinary capacity for mathematics. This astonishes me greatly, for, after his examination, I believed him to have but little intelligence. - (Later in life he was sent to prison twice.)

I am sure that you remember at least one of them, I am not sure about the other though. Do you recognize the persons already? - One of them is recognized as one of the greatest minds that ever lived. The other has been criticized for the lack of depth in his work.

Who are they?
- Mystery person #1.
- Mystery person #2.

Wednesday, February 22, 2012

Abstract Algebra E-222 video 26 Rings 2


Watched lecture 26 of the Harvard Abstract Algebra series. - What can you say about the complex number $z$ if $(2+i)z$ must be an integer?

Prof. Gross ... "ideals in the Gaussian integers $\mathbf{Z}\left[i\right]$ of type $\mathbf{Z}/p\mathbf{Z}$".

These lectures were recorded in 2003 and are basically saved for all generations to come. Imagine that the lectures of Gauss were recorded on video! Euler and Gauss will be remembered forever by their name and picture, but the great mathematicians of today and tomorrow will be remembered by their video lectures.

Monday, February 20, 2012

TeXnicCenter 2.0 Alpha 4

TeXnicCenter have released Alpha 4 of TeXnicCenter, the LaTeX IDE for Windows.

More at:

Abstract Algebra E-222 video 25 Rings 2

Every word counts in mathematics.

Every non zero single-variable polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted up to its multiplicity. ( Fundamental theorem of algebra, Wikipedia )

The polynomial $x^2-1=0$ has ( thus ) two roots: $(1, -1)$. However, if we consider the coefficients of the polynomial as elements of the ring $\mathbf{Z/8Z}$ then the polynomial has four roots: $(1, -1, 3, -3)$.

$x^2-1$ has $4$ roots...

In lecture 25 professor Gross explains how the division and Euclidean Algorithm can be applied to polynomials in Polynomial Rings over a field.

Sunday, February 19, 2012

The squared wheel


The squared wheel is a beautiful example of mathematical thinking: thinking out-of-the-box, generating alternatives and cutting one's way through the jungle. It is also a motivator to think deeply about a problem, even if it seems obvious.

If you have asked yourself the question: "Are there more wheel configurations possible than just the circle and the square?" then, I suppose, you got what it takes. ;-)

Have a look at the Wolfram demo 'Roads and Wheels'.

Saturday, February 18, 2012

Abstract Algebra E-222 video 24 Rings 1


Just watched a video where Benedict Gross introduces Ring Theory. I don't think you can learn Ring Theory ( or any mathematics for that matter ) by just watching a video.

In the business of commercial education, in programming for example, teachers are often confronted with students ( sent by their employers ) who expect to leave as a qualified programmer just by hanging in their chairs during the course. Needless to say they leave as empty headed as they came in.

But if you watch prepared you can pick up a lot from this professor. In this first lecture he explains why there is such a field as Ring Theory in the first place. Where did it come from? And most of all: what are the important topics we have to watch in this field? ( I.e. Ideals and Unit Groups ). You may wonder why I gave this post the M336 ( Groups and Geometry ) hash-tag, it is because Rings and abelian Groups ( and Number Theory ) are intimately connected and one of the objectives of M336 is the classification of all abelian groups. - By the way, the word is abelian group and not Abelian group despite the fact that the word abelian comes from Niels Abel. Writing a name lowercase is the highest possible honor in mathematics. ( So I have been told... ).

$(\mathbf{Z/nZ})^{\times}$ has $\phi(n)$ elements

At the end of this lecture he mentions that Group Theory is a really hard subject and all that. The thing with Group Theory is that it has to sink in quite a while before it clicks and opens up to you.

Planet Math servers crashed

From Planet Math:

After many months of instability culminating in a major crash, we have decided to imply migrate to the next version of our content management software, Noosphere 1.5. As of now all data since late 2011-10-23 has been lost, and some functionality is broken or missing. Please be patient and report problems.

I sincerely hope PlanetMath survives this setback. Oh, and Noospere 1.5 looks great!

Friday, February 17, 2012

Polynomial exercise - Solution.

In a recent post I proposed the following exercise.

Let $$x^3 + bx^2 + cx + d$$ be a polynomial with coefficients in $\mathbf{Q}$. We ask which condition(s) $b,c,d$ must satisfy in order that one ( any ) root be the average of the other two roots?


Let the roots of $x^3 + bx^2 + cx + d = 0$ be $\alpha_1, \alpha_2, \alpha_3$. Clearly, the roots must satisfy the following equations:
\alpha_1=& \frac{1}{2}\alpha_2 + \frac{1}{2}\alpha_3 \\
\alpha_2=& \frac{1}{2}\alpha_1 + \frac{1}{2}\alpha_3 \\
\alpha_3=& \frac{1}{2}\alpha_1 + \frac{1}{2}\alpha_2 \\
This is a rank $2$ system of linear equations with solution $( \alpha_1, \alpha_2, \alpha_3) = \lambda (1,1,1) $, and implies that the equation has three equal roots and can thus be written as follows:$$(x-\alpha)^3 = x^3-3\alpha x^2 + 3\alpha^2 x - \alpha^3$$.
So if the root is $\alpha$ then $b=-3\alpha, c=3\alpha^2$ and $d=-\alpha^3$.

Thursday, February 16, 2012

Drawing ( friezes ) with Bezier curves (2)

#OpenUniversity #M336

Of, course colors can be added and so forth which may generate some interesting problems for M336 GE1 Counting with Groups.

See also:

- 'Frieze group pmm2'
- Drawing ( friezes ) with Bezier curves

Wednesday, February 15, 2012


I am studying some more about polynomials, the topic of symmetric polynomials for example, is an interesting one.

Let $$x^3 + bx^2 + cx + d$$ be a polynomial with coefficients in $\mathbf{Q}$. We ask...

...which condition(s) $b,c,d$ must satisfy in order that one ( any ) root be the average of the other two roots?

To be continued.

Drawing ( friezes ) with Bezier curves

#OpenUniversity #M336

Although the friezes in post 'Frieze group pmm2' consisted of only straight lines it is possible to create friezes containing Bezier curves in Mathematica using the BezierCurve function.

The following frieze will look familiar to M336 students. ;-)

Sol Lederman is "Inspired by Math"

Sol Lederman announced a series of podcasted interviews...

...with people who are inspired by math and how they're helping others to be inspired.

The first interview with Keith Devlin ( Senior Researcher at CSLI ) is available here.

Tuesday, February 14, 2012

Frieze group pmm2

#OpenUniversity little success story

The most common barrier to effective study is the so called 'Lack of Mass' ( Applied Scholastics ). A subject has not enough mass -for you- when you don't like it, aren't interested in it, can't see the purpose of studying it, etc.

If this situation occurs then you simply (...) have to 'add mass'. I did it for Open University course M336 IB3 Frieze Patterns by programming a frieze pattern tool in Mathematica. I like programming and if you can program a topic it is proof that you understand the topic. Now friezes live for me. I know them all, including the recognition algorithm.

Here are some applications of the tool I made.

A graphical proof that a frieze containing the letter H ( i.e. HHH... ) has symmetry group pmm2.

Or do it the other way around: take a letter R frieze and transform it to a frieze with p1a1 symmetry.

And now I can't wait to start with the Wallpaper Patterns. So, if you don't like a subject you can do two things: wait until you start liking it which may be never, or take creative action so that you -do- like it.

Sunday, February 12, 2012

Greatest Common Divisor - GCD

Jan van der Meiden created a graph that shows if the GCD of an integer pair is larger than one by coding the intersection dark versus white when the integers are relatively prime.

Through this graph shimmer numerous patterns.

Have a look here at his site:

Saturday, February 11, 2012

Mathematics and culture


Because I study mathematics at a British university I am starting to notice that the British insist on doing things their -own- way as much in mathematics as they do in general. I am not judging this, but it fits the pattern, i.e. traffic, currency, etc.

It is just that I thought mathematicians would be -wiser-. Wiser, how ignorant of me, how can I possibly understand the essence of 'being British'? I can't, of course.

Let me give two ( recent ) examples.

An Open University forum moderator switching to UPPERCASE in reply to my mentioning that permutations are applied from left to right in default GAP while the only correct way is the British right-to-left. ( Going uppercase is about the rudest imaginable attitude known in internet etiquette and as you can imagine I was flabbergasted. )

A note in a book published by the American Mathematical Society written by an English mathematician. I quote:

"Readers who prefer this convention should read this book upside down in a mirror."

He referred to a generally accepted style of notation in continental Europe. My jaws dropped. This wasn't meant as a joke.

;-) !

2012 Alan Turing Year

The year 2012 is Alan Turing year, Turing was born on the 23rd of June in 1912. And of course, because of:
"There isn't a discipline in science that Turing has not had an impact upon."
Considering the 20th century was the year that computers were 'born', which mathematician will be remembered as the foremost one of that century, in say 300 or more years? What I particularly like about Alan Turing is that his papers are so -accessible-. By an undergraduate, at least. Turing was the first programmer and the first hacker: he cracked the German Enigma which changed the course of WW2. The story of his life reads like a thriller, a film noir perhaps.

More, in the course of 2012.

Turing Year on Twitter.

Wednesday, February 8, 2012

Riemann Hypothesis ( Video )

Very well done video about the history, and the implications of the RH.

The paradox of abundance

The number of people connected to the Internet is still growing. Is Planet Earth becoming like The Borg in Star Trek? Technology is changing the world into a better place. I am not sure if I agree though. Have to think about it.

Tuesday, February 7, 2012


I have missed out on a game, I suppose. The NeoCube. Must-have for math(s)-geeks, if you ask me. I don't watch television and I don't go out shopping anymore so it's rather obvious that I missed it. But I haven't seen any people playing with it either. Shopping, buying books, I loved it. The Internet took that away. What I do like about Internet is that people bring their laptop to the cafe. Have coffee while you read the online papers, that's nice. - Anyway, this is the NeoCube I mentioned.

Symmetry, symmetry!

[ Housekeeping ] Comment Moderation = ON

I have switched comment moderation back to on. Switching it off hasn't worked, it made things worse. I keep forgetting to check if there are comments to reply too and worse: very odd comments got through. Like: "What an interesting point on group theory you just made. Joe" Where Joe supposedly was a link to a profile but instead it jumped to a gambling site, or worse.

Sunday, February 5, 2012

Friday, February 3, 2012

Music inspired by mathematics

Mathematical songs

I sincerely love this song, Pi. The image is taken from the movie Pi directed by Darren Aronofsky ( Black Swan, Requiem for a Dream ). ( The performer did not take credit for the lyrics. )

More serious, the lyrics on Kraftwerk's, ( visionary ) Computer World album, created in the early 1980s, described a world dominated by computers, like today. Kraftwerk had a major influence on the music of the 80s and 90s. The lyrics of Numbers, one of the songs on Computer World were abstract and mathematical. The video is recent ( 2010 ) and of HD quality. Enjoy!

Crystal Pite

It is not a coincidence that scientists and artists flock together. From the moment they become students they meet and interact. Scientists need to go off the main roads to find new undiscovered territory. In a sense artists guide them because they help them to think differently. Likewise artists are influenced by science. ( Embedding has been disabled but the interview is just one click away ) Crystal Pite talks about Frontier. A ballet choreographer speaks about dark matter, dark energy and the line between what we know and what we don't know.

Thursday, February 2, 2012

The integers and the natural numbers have the same cardinality

One could argue that there are twice as many integers as there are natural numbers since for every natural number there are two integers: $1 \mapsto (1,-1)$, $2 \mapsto (2,-2)$.

You can imagine that there was quite some opposition from within the mathematics community when Georg Cantor (1845-1918) proposed the following theorem:
Two sets A and B have the same cardinality if there exists a bijection, that is, an injective and surjective function, from A to B.

A sequence for $\mathbf{N}$ is $$s(n) = \sum_{k=1}^n 1 \ ,$$
$$ \begin{array}{cc}
n & s(n) \\
1 & 1 \\
2 & 2 \\
3 & 3 \\
4 & 4 \\
\cdots & \cdots \end{array} $$

Likewise a sequence for $\mathbf{Z}$ is: $$s(n) = \sum_{k=1}^n (-1)^{k+1} \cdot k \ ,$$
$$ \begin{array}{cc}
n & s(n) \\
1 & 1 \\
2 & -1 \\
3 & 2 \\
4 & -2 \\
\cdots & \cdots \end{array} \, $$ We can thus establish a bijective (one-to-one) map between $\mathbf{N}$ and $\mathbf{Z}$.

By the theorem above we can conclude that $\mathbf{N}$ and $\mathbf{Z}$ have the same cardinality ( 'number of elements' ).

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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)