Walking Randomly

The 61st Carnival of Mathematics will be hosted here on 1st January 2010 so please head over to the carnival submissions page and help make 2010 start with a mathematical bang.  For an idea of what the carnivals look like check out the 60th over at SumIdiots blog and the 59th over at The Number Warrior.

Picture the scene….

A math instructor and her student are working through a problem together and on the first line it says

f(x) = 2(x+3)

“So” says the instructor “What do we have here?”

The student thinks for a second before replying, “f of x equals 2 of x+3”

The instructor gives a long sigh and puts her head in her hands.

“Have I taught you nothing?” she asks desparingly as the student looks on in bewilderment. Who has the conceptual misunderstanding of this situation; the teacher or the student?

3435 = 3³ + 4⁴ + 3³ + 5⁵

Isn’t this a lovely little identity? Check out this paper by Daan van Berkel in the arXiv for the details.

Earlier today I was chatting to a lecturer over coffee about various mathematical packages that he might use for an upcoming Masters course  (note – offer me food or drink and I’m happy to talk about pretty much anything). He was mainly interested in Mathematica and so we spent most of our time discussing that but it is part of my job to make sure that he considers all of the alternatives – both commercial and open source. The course he was planning on running (which I’ll keep to myself out of respect for his confidentiality) was definitely a good fit for Mathematica but I felt that SAGE might suite him nicely as well.

“Does it have nice, interactive functionality like Mathematica’s Manipulate function?” he asked

Oh yes! Here is a toy example that I coded up in about the same amount of time that it took to write the introductory paragraph above (but hopefully it has no mistakes). With just a bit of effort pretty much anyone can make fully interactive mathematical demonstrations using completely free software. For more examples of SAGE’s interactive functionality check out their wiki.

Here’s the code:

def ftermSquare(n):
 return(1/n*sin(n*x*pi/3))

def ftermSawtooth(n):
 return(1/n*sin(n*x*pi/3))

def ftermParabola(n):
 return((-1)^n/n^2 * cos(n*x))

def fseriesSquare(n):
 return(4/pi*sum(ftermSquare(i) for i in range (1,2*n,2)))

def fseriesSawtooth(n):
 return(1/2-1/pi*sum(ftermSawtooth(i) for i in range (1,n)))

def fseriesParabola(n):
 return(pi^2/3 + 4*sum(ftermParabola(i) for i in range(1,n)))

@interact
def plotFourier(n=slider(1, 30,1,10,'Number of terms')
,plotpoints=('Value of plot_points',[100,500,1000]),Function=['Saw Tooth','Square Wave','Periodic Parabola']):
    if Function=='Saw Tooth':
     show(plot(fseriesSawtooth(n),x,-6,6,plot_points=plotpoints))
    if Function=='Square Wave':
     show(plot(fseriesSquare(n),x,-6,6,plot_points=plotpoints))
    if Function=='Periodic Parabola':
     show(plot(fseriesParabola(n),x,-6,6,plot_points=plotpoints))

Within every field of human endeavor there is a collection of Holy Grail-like accomplishments; achievements so great that the attainment of any one of them would instantly guarantee a place in the history books. In Physics, for example, there is the discovery of the Higgs boson or room temperature superconductivity; Medicine has the cure for cancer and how to control the ageing process whereas Astronomy has dark matter and the discovery of extra-solar planets that are capable of supporting life.

Of course, mathematics is no exception and its Holy Grail set of exploits include problems such as the proof of the Riemann hypothesis, the Goldbach conjecture and the Collatz Problem.  In the year 2000, The Clay Mathematics Institute chose seven of what it considered to be the most important unsolved problems in mathematics and offered $1 million for the solution of any one of them.  These problems have since been referred to as The Millenium Prize Problems and many mathematicians thought that it would take several decades before even one was solved.

Just three years later, Grigori Perelman solved the first of them –  The Poincaré conjecture.  Stated over 100 years ago (in 1904 to be exact) by Henri Poincaré, the conjecture says that ‘Every simply connected closed three-manifold is homeomorphic to the three-sphere’.  If, like me, you struggle to understand what that actually means then an alternative statement provided by Wolfram Alpha might help – ‘The three-sphere is the only type of bounded three-dimensional space possible that contains no holes.’  99 years later, after generations of mathematicians tried and failed to prove this statement, Perelman published a proof on the Internet that has since been verified as correct by several teams of mathematicians.  For this work he was awarded the Field’s Medal, one of the highest awards in mathematics, which he refused.  According to Wolfram Alpha, Perelman also refused the $1 million from the Clay Institute but, as far as I know at least, he has not yet been offered it (can anyone shed light on this matter?).

Yep, Girgori Perelman is clearly rather different from most of us.  Not only is he obviously one of the most gifted mathematicians in the world but he also sees awards such as the Field’s Medal very differently from many of us (after all, would YOU refuse such an award – I know I wouldn’t!).  So, what kind of a man is he?  How did he become so good at mathematics and why did he turn down such prestigious prizes?

In her book, Perfect Rigor, Masha Gessen attempts to answer these questions and more besides by writing a biography of Perelman.  Starting before he was even born, Gessen tells Perelman’s story in the words of those who know him best – his friends, colleagues and competitors.  Unfortunately, we never get to hear from the man himself because he cut off all communications with journalists before Gessen started researching the book.  Despite this handicap, I think that she has done an admirable job and by the end of it I have a feeling that I understand Perelman and his motives a little better than before.

This is not a book about mathematics, it is a book about people who DO mathematics and gives an insight into the pressures, joys and politics that surround the subject along with what it was like to be a Jewish mathematician in Soviet-era Russia.  What’s more, it is absolutely fascinating and I devoured it in just a few commutes to and from work.  With hardly an equation in sight, you don’t need pencil and paper to follow the story (unlike many of the books I read on mathematics), all you need is a few hours and somewhere to relax.

The only problem I have with this book is that by the end of it I didn’t feel like I knew much more about the Poincaré conjecture itself despite getting to know its conqueror a whole lot more.  Since I didn’t know much (Oh Ok…anything) about it to start with this is a bit of a shame.  Near the end of writing this review, I took a look at what reviewers on Amazon.com thought of it and it seems that some of them are also disappointed at the mathematical content of the book and they come from a position of some authority on the subject.  The best I can say is that if you want to learn about the Poincaré conjecture then this probably isn’t the book for you.

If, on the other hand, you want to learn more about the human being who slayed one of the most difficult mathematical problems of the millennium then I recommend this book wholeheartedly.

The carnival is back and it’s better than ever!

The 59th Carnival of Math has been published by Jason Dyer over at Number Warrior and is truly epic!  The range of topics on offer is astonishing including mathematical clocks, ants on rubber bands, an entire quiz on the number zero, the game of life, cheating the derivatives market and whether or not an omnipotent being can create a rock so heavy that He/She cannot lift it?

The next carnival (the 60th)  is set to be published on Friday December 4th but currently has no host set.  If you are a math blogger and would like to get involved then drop me a comment and let me know.  Hosting a carnival is not difficult and is a great way to get involved with the wider mathematics blogging community.  As a bonus, it will help get your blog a lot of free publicity!

Ideally, I’m looking for someone who has never hosted a carnival before but old hands are welcome to volunteer too.

If you simply can’t wait for December’s Carnival of Maths then you can get your fix of community mathematical  goodness from the next Math Teachers at Play Carnvial.  Set to be hosted on 20th November by Denise of Let’s Play Math, the MTAP carnival focuses on preK-12 mathematics but, despite its name, it isn’t just for math teachers – anyone can join in.

Update (9th November 2009): The quest for the next carnival host is over.  The 60th edition of the Carnival of Math will be hosted on Friday December 4th over at Sumidiot.  Start getting your submissions in now and help make this the best carnival ever.

Back in August 2008, Joel Spolsky and Jeff Atwood launched a question and answer site aimed squarely at programmers and it has been a massive hit.  Called Stack Overflow, the site currently contains over 330,000 questions and answers covering pretty much every aspect of programming you care to mention from What IDE to use for Python through to smart pointers in C++ and everything in between.  With a userbase running into the thousands, you can be sure that someone, somewhere will know the answer to your programming question no matter how obscure.  All this and yet, in true internet tradition, it won’t cost you a penny.

Well, time moves on and Stack Overflow spawned other sites based on the same web technology.  Server Fault, for example, is aimed at system administrators and finally, to complete the trilogy, we have Super User which is geared towards general computer enthusiasts.  Although neither of these have been as popular as the original programming site (Server Fault currently has 17,000 questions and Super User has just over 12,000) they are still very useful resources.  More importantly, they demonstrate that you can use the Stack Overflow technology to build any community question and answer site.

I guess it was inevitable that someone would eventually build such a site for mathematics.  Math Overflow is currently in the beta stage of development and only contains just under 400 questions at the time of writing but it has potential.  It just needs a bigger audience, so head over there and get questioning.

I guess you could think of it as a people-powered version of Wolfram Alpha!

In my previous post I asked how people felt about the future of the Carnival of Maths since it had become poorly organised lately.  Well, a group of us got together, rolled up our sleeves and sorted something out.  The result should see the Carnival going for a couple more years yet at least.  Here are the measures we have taken

• I have taken over the running of the main carnival submission page and have updated it.
• I have created a Twitter feed for the carnival.
• Rod Carvalho has created a google group for people who have contributed to the carnival over the years.  A group of us have been discussing the carnival’s future there and, between us, we will be able to ensure that it stays well organised in the future.  As far as I am concerned, this group runs the show and if they vote to have me replaced as carnival co-ordinator at any time for any reason then I will hand over the reigns.
• The Carnival of Math is now monthly and will be published on the first Friday of every month.  The next one will be on Friday 6th November and will be over at Jason Dyer’s Number Warrior.
• From November, The Math Teachers at Play Carnival will be running on the 3rd Friday of every month in order to co-ordinate with the new Carnival of Math schedule.  This means that you are never far away from your next fix of mathematics blog posts.

That’s pretty much it.  Thanks to everyone who has contributed to the disucssion and to those who helped set things up.  Here’s to the next carnival :)

The Carnival of maths has been around for a while now, recently reaching it’s 58th edition, but its organisation has become erratic of late. The first carnival was put together by Alon Levy who organised it ever since, continuing even after he officially hung up his blogging hat. The mathematics blogging community owes Alon a big debt of gratitude so I hope you all don’t mind if I say “Thank You” to Alon on all our behalves.

Recently, however, it seems that Alon’s other life commitments have caught up with him since the main Carnival page has not been updated since Februrary and the Carnival submission form indicates that the carnival has closed.  I’ve also tried to email him a couple of times (before I hosted the 58th carnival for example) but as yet have received no response.  Alon – if you are reading this then I hope you are doing OK.

So, onto the question posed in this post’s title.  What should the future hold for the Carnival of Maths?  I have thought of a few possible options as follows

• Let it end here.  The community has Math Teachers at Play and that may be enough carnival for us all.
• Continue the carnival in it’s current form but co-ordinated by someone else.
• Start a new carnival series with a title like ‘New Carnival of Maths’.
• None of the above?  Your suggestions would be welcome.

Personally, I favour option 2 – to continue the carnival in its present from but to reduce posting frequency to once a month.  As for co-ordinator – I would be happy to do it if there were no objections (and host a co-ordinating page here) – but if it is felt that this is inappropriate then I would happily step aside and let someone else do it.

It’s your carnival so what do you think?  Comments welcomed.

Among my friends I am the resident mathematics nerd and so I get asked all sorts of weird and wonderful little questions ranging from the trivial to the impossible to answer.  One such question that came up while on a night out was ‘why is a minus times a minus equal to a plus?’

I have to admit that I couldn’t come up with an answer that satisfied my non-mathematical friend which bugged me no end so I asked around….’How would you explain it?’. I asked on Twitter, I asked in my office, I asked some math teachers I know. I even came up with a ‘proof’ (I am an ex-physicist so I apologise if this is considered to be far from rigorous) as follows.

From the distributive law of multiplication we have

a*(b+c) = a*b + a*c

which holds for all real numbers. So it will hold if we put a=-1, b=1 and c=-1 right? let’s do it!

(-1)*(1+(-1)) = (-1)*1 + (-1)*(-1)

1+(-1) = 0 and -1*1 = -1 so we get

(-1)*0 = -1 + (-1)*(-1)
0 = -1 + (-1)*(-1)

rearrange to get

1 = (-1)*(-1)

Simple enough but not exactly intuitive is it? By the way, would any proper mathematicians care to comment as to whether this constitutes a proof or not?

Ideally I wanted an explanation that was completely intuitive and so far the best I have come across is the explanation from my friend Paul over at Crossed Streams who’s fiancee asked us this very question while we were all on a night out.  Other good explanations involved ideas such as vectors and direction but when we tried those out on our target we were met with suspicion – ‘I thought we were talking about numbers and you explain using vectors!’

What fascinated me was that my tweets asking for explanations were possibly the most popular I have ever made. I had people phone me at my office to ask ‘So did anyone come up with a good explanation then?’

A minus times a minus is a plus….everyone knows it….but how would you explain it such that even the least mathematical person would grok it?