I’ve been working on other things for a while but have started to get moving on my latest project: Sequences and Series. This began with the leftovers from How to Think Like a Mathematician. I had written quite a few chapters on analysis, in particular, on the definition of the limit of a sequence. After all, the concept of epsilon-delta proof is a key topic to grasp in mathematics and require deep logical thinking for most students. Anyhow, the chapters had to go as the book would have been too long. I had intended to use them as a basis for a book on real analysis but got involved in a book on complex analysis. Next, one of my colleagues produced a book on real analysis! So rather than be ungentlemanly direct competition for him I’ve used my notes to create a website on Sequences and Series. It’s a bit basic at the moment (and has been up for a while) but I’ll add more this year. A book will follow in due...

## Gresham/LMS Lecture

posted by Kevin Houston

This week the joint Gresham College and London Mathematical Society lecture will take place. Reidun Twarock of the University of York will give a talk Geometry: A New Weapon in the Fight Against Viruses. The details are on the Gresham College website. In case you can’t make the talk on Wednesday, then maybe there is a MathsJam near you on Tuesday. And if you can’t make either, then maybe you would like to see last year’s Gresham/LMS talk by Marcus du...

## Nate Silver on Panorama

posted by Kevin Houston

I’m not a regular watcher of Panorama. It is a TV programme that has dumbed down considerably in my lifetime and I have probably only seen two programmes in the past decade. One was a couple of weeks ago on male suicide and the other was this Tuesday’s episode about Nate Silver attempting to predict the outcome of the forthcoming General Election here in the UK. Nate Silver’s book The Signal and the Noise, on the uses of statistical methods for prediction, is one of my favourite books of recent years and is well worth reading if you haven’t read it. This programme is not so good. Silver was successful in predicting, against a backdrop of pundits who said it was too close to call, that Obama would win the last US Presidential election. Hence, it is natural to see if he can do the same in a UK General Election. My recollection is that he discusses this in his book and points how it is much much harder it is. Given recent developments in the UK it is now even harder… Anyhow, the programme itself is rather lightweight. The presenter Richard Bacon takes Silver round the country explaining the UK system, Silver goes back to US, and makes a prediction. There is no explanation of the statistical methods involved or how they relate to the election. How Silver arrives at the conclusion is overlooked. I’m not expecting an explanation of Bayesian inference or some such on prime time TV but the probabilities and ranges in his prediction have been removed and replaced with absolute numbers. An averagely intelligent viewer can grasp the concept of range surely. Hence, no major insight is gained. The prediction is not that different to what many would guess. My prediction was that, due to soft UKIP support, the Conservatives would be largest party but no majority, and that Labour and SNP would have a combined majority. And, apart from the numbers in the centre of the ranges, that’s all that the programme gives us. That’s a bit disappointing. The programme is available for 11 months on iPlayer but I guess that it is not available in all territories. Watch the programme here. Silver’s report, made in conjunction with three British academics, is available on Silver’s site The Five Thirty Eight. The three academics, Chris Hanretty, Benjamin Lauderdale, and Nick Vivyan, maintain the site electionforecast.co.uk. On the subject of election results, why do newspapers never give us the number of Don’t Knows when they publish a poll result before the election? I would say that group is important. Also, the number of Don’t Knows can be as large as 40% at the start of a campaign. It would be interesting to see how this changes during an election and to know why they are Don’t Knows. Just had to get that grumble off my chest… UPDATE: Well, just about everyone got it wrong. The Conservatives achieved a majority government! David Spiegelhalter has quite a bit to say about the polling problems on his Understanding Uncertainty...

## Eduardo Sáenz de Cabezón – Math is Forever...

posted by Kevin Houston

It has been a while since I posted a TED talk. There haven’t been many good ones in recent years. I enjoyed the one below (though I think he could have made clear that men respond the same way as women after asking the “What do you do?” question!). Further Reading The paper on the Honeycomb Conjecture by Thomas Hales starts off with a very good introduction before getting down to the higher level mathematics. Pappus of Alexandria Freeform Honeycomb Structures Not directly related but is an interesting paper from a conference I was at last...

## Popular Lectures of the London Mathematical Society...

posted by Kevin Houston

The line up for this year’s London Mathematical Society Popular Lectures has finally been announced. As this year marks the 150th year of the LMS there are four lecturers instead of the usual two. We have Professor Martin Hairer, FRS – University of Warwick (and recipient of a Fields Medal last year) Professor Ben Green, FRS – University of Oxford Dr Ruth King – University of St Andrews Dr Hannah Fry – University College London The lectures will be at different times and different places, see the Popular Lectures webpage. Last year’s lectures are available...

## 80s rubric

posted by Kevin Houston

The last post was a monster post. This one is a bit shorter as it exam time at my university so I’m a bit busy. Students these days have it easy with regards to exams. They are given far too much information about how to get good marks. When I was a lad we had to guess what an examiner wanted. Just look at the rubric of this 1980s exam (click to enlarge): I just love “Full marks may be obtained for complete answers to about FIVE questions”! And this is not an isolated case, I have a stack of exam papers like...

## The Beatles’ Magical Mystery Chord

posted by Kevin Houston

On the 16th April 1964 the Beatles, George Harrison, Paul McCartney, John Lennon and Ringo Starr, were on the cusp between stardom and mega-stardom. Groundbreaking albums, hit films, cartoons, royal honours, Sgt. Pepper, Indian mysticism, and an acrimonious break up all lay in their future as did the murder of Lennon and attempted murder of Harrison by separate mentally disturbed fans. That was all to come. Their task that night was to record an impressive opening track for their forthcoming debut film and the album to accompany it. Their producer, George Martin, wanted something spectacular: ‘We were looking for something big to open it with, an introduction. It needed a strong chord, a dramatic thing’ (p487 of Dominic Pedler, The Songwriting Secrets of the Beatles, Omnibus Press, 2003). Those present in the Abbey Road studio to record Lennon’s song A Hard Day’s Night could not have imagined that 50 years later the events would still be analyzed and dissected. Particularly since the focus of the analysis is almost not a piece of music, it’s a short sound, less than 3 seconds long, a crashing, ringing, chiming sound that has caused arguments and discussions between Beatles’ fans and musicologists ever since it was recorded. The noise is impossible to describe accurately in words – the famous quote ‘writing about music is like dancing about architecture’ comes to mind. The sound can be heard here: http://www.kevinhouston.net/blog/wp-content/uploads/2014/12/original-chord3.wav The central question is simple: What is it? That is, what notes are played and who is playing them? Many versions have been suggested. In his massive Beatles book, The Songwriting Secrets of the Beatles, Dominic Pedler collects twenty one educated guesses from various sources and devotes over 40 pages to discussion, including his own theory. It is not difficult to produce a chord that is close – strumming a guitar without fretting produces a similar sound. It’s close. But close is not exact, right? So, what is it really? In 2004 a mathematician claimed to have discovered this musical holy grail by applying mathematics. Once and for all, the riddle was solved because, after all, mathematics is not wrong and you can’t beat the scientists with their fancy abstract toys. Except there was a problem. He got it horribly wrong. Here, for the honour of all mathematicians, I would like to put the record straight — or at least straighter. The mathematical tale of the Beatles’ Magical Mystery Chord is a tale of 18th century mathematicians, the study of heat, Karaoke tricks and a measure of luck. The quest begins I am a mathematics lecturer who enjoys promoting mathematics to school children and the general public. My introduction to the mystery of the chord came not from a love of the Beatles music but a desire to show off in front of my parents. My mathematics promotion involves giving talks all over the country but as I live in the north of England and my parents live in a secluded part of the north east of Scotland, they would under normal circumstances be unable to see their second son on a stage explaining mathematics. Back in 2011 I heard that the British Science Festival would be held the following year in Aberdeen which is close to where my parents live. The festival is held in a different city in the UK each year and aims to engage the general public in science. Hundreds of events take place over a week in September with debates, demonstrations and hands-on exhibitions at the local university as well as theatres and even, like maths busking, in the street. (Maths busking is as it sounds. Mathematicians go out in the street and do mathematics to entertain passers-by. I’ve tried it and it is hard. See my report here.) All I had to do was offer to give a talk, get accepted and I would get my chance to impress...

## How to get a good degree 3: How to become a straight-A student...

posted by Kevin Houston

Usually I am against books offering general study advice, I favour those that focus on a particular subject. (Which is why I wrote How to Think Like a Mathematician. Mathematics students may also be interested in Lara Alcock’s How to Study for a Mathematics Degree.) I’ll make an exception for Cal Newport’s How to Become a Straight-A Student. Let’s get the bad stuff out of the way first. The subtitle is The Unconventional Strategies Real College Students Use to Score High While Studying Less which, along with the book’s other marketing, makes the scam-like sounding promise that you can do less work and get better grades. This can of course happen but does lend the book a feeling of too-goo-too-be-true. Furthermore, the book could do with some trimming of excess material though, at 216 pages, it is quite short for this type of book. The book could do without the regular mentions of partying and beer swigging but I suspect I’m not the target market for those bits. But leave aside those problems. Why would I recommend the book? Essentially, most of the advice is good. There is the standard stuff that all students know that they should do: get plenty of rest, eat properly, do your work in the morning between lectures to gain a sense of accomplishment. The non-standard stuff is good too. He doesn’t advocate lots of highlighter pens or even a highly detailed to do list. Instead he mostly focusses on the methods for efficient and effective learning. For example, on page 105, he talks of the Quiz-and-Recall method, [emphasis mine] Whether it’s philosophy or calculus, the most effective way to imprint a concept is to first review it and then try to explain it, unaided, in your own words....

## Summer Hiatus…

posted by Kevin Houston

My unscheduled summer hiatus from blogging arose from a family emergency that meant I was in stuck in Brazil for about 5 weeks. Not a bad place to be stuck but under the circumstances it was difficult to find time to finish my planned summer posts. For example, I had planned to post about the Fields Medals and as it happened lost the opportunity to report from Brazil about a Brazilian winner. The short story is that, to me, the Brazilians seemed mostly proud but surprised that a Brazilian was even in the running. The hiatus also meant that the deadline for my next textbook wooshed by. Ok, in reality I was never going to make it but I’m even further behind than expected. Anyhow, service will resume when I get back on track after all the time lost. For instance, I’ll let you know what the book is...

## LMS Popular Lectures

posted by Kevin Houston

The London Mathematical Society runs a regular Popular Lectures Series. These are mathematics lectures by (usually) research mathematicians. In recent years a pair of lecturers has performed in London and Birmingham. This year they will be given by Julia Gog and Kevin Buzzard. (The London performance was last night and the Birmingham one is in September.) If you haven’t got tickets, then the good news is that the lectures are recorded and put online. Currently, an admittedly incomplete, collection of previous lectures is hosted in two different places: Lectures from 2008-2013 Lectures from 1986-1996 Previous speakers have included Sir Tim Gowers, Sir Roger Penrose, Reidun Twarock, Matt Parker, Mark Miodownik, Ray Hill, Vicky Neale and Dorothy...

## Another Hard Day’s Night...

posted by Kevin Houston

I haven’t posted in a while for many reasons. One of which is that I’ve been trying to finish a textbook. My target date is July 31 to have the final draft finished. I’m not sure I’m going to make it… Anyhow, I noticed that the Beatles’ first film A Hard Day’s Night is being rereleased. ‘But what’s the opening chord to the title song?’ I hear you say. Funny you should ask, that gives me a chance to show again my modest attempt at solving the...

## Persi Diaconis Lecture on Martin Gardner at BMC...

posted by Kevin Houston

The name Martin Gardner is familiar to most mathematicians. He wrote numerous on mathematics from a culture and leisure viewpoint. (You can find his books on Amazon.) Next week Persi Diaconis will give a talk at the British Mathematical Colloquium (BMC) on the life and work of Martin Gardner. The BMC is an annual gathering of research mathematicians in the UK and beyond. Diaconis’ talk is a public lecture so anyone may attend but a (free) ticket is required. Details of the talk are here. I’ll be attending so do say hello if you see me. For all of those unable to attend but want to know a bit more about Gardner then Diaconis has co-written a biography of Gardner (as well as a great mathematical magic book). There is also a recent article in the New York...

## Mathematics of Love

posted by Kevin Houston

Today is St. Valentine’s Day, the day in much of Western culture arbitrarily designated to be the day for love. So let’s see what mathematics has to say on the subject. Finding a relationship with someone special is often about being introduced to people and sifting out the inappropriate. It seems clear that there is plenty of scope for the application of statistical techniques to the processes of meeting and weeding. First up is a great video from the ever-so-slightly geeky Amy Webb, who used mathematics to calculate the odds of finding a mate in Philadelphia. After producing a figure of 35 suitable men satisfying her criteria in a city of 1.5 million people she realized that she would have to turn to maths for help. There’s even a book, Data, a Love Story. Over on Wired, Chris McKinlay’s attempts to hack OKCupid’s online dating service is profiled. The article left me in two minds. Is this is great use of mathematics or is it just a bit creepy? Seemingly, the approach worked for him and no one is reported injured, so perhaps I shouldn’t judge. As a bonus, Wired also produced a handy infographic slideshow describing tips for improving online dating profiles. Top tips: Avoid Karaoke, get into surfing. My favourite though is that it is more attractive to mention “cats” than “my cats”. If everything goes well with the dating, then how do you arrange the wedding? “With maths” is not the standard answer. The Guardian reports on a statistically modelled wedding. This solves the centuries old problem of how to write the guest list. After all, you don’t want too few guests or too many accepting. Next post: Calculating the likelihood of...

## Rafael Araujo – A New Escher?...

posted by Kevin Houston

One needs only look at some of Fomenko’s monstrosities to see that mathematics and art don’t always mix well. Exceptions are rare with Escher the benchmark for excellence. A recent Wired article on the Venezuelan artist, Rafael Araujo, describes how he produces his mathematically constructed pictures. I can see these joining Escher’s pictures on the walls of mathematicians throughout the world over the next few years. The full article is here but you can also see stunning examples on his...

## Hilbert Hotel Video

posted by Kevin Houston

Belated Happy New Year! Here’s another of those short educational TED videos. This time on the Hilbert Hotel. (A cartoon Hilbert does seem to make an appearance but it is hard to tell as he isn’t wearing the hat. Do you think Hilbert wore that hat just once and had the misfortune of having it appear in his most famous picture?) Anyhow: You can find out more at the TED Education page for the video. Oh, and this week is the last chance to win a signed copy of Simon Singh’s book on The Simpsons! See...

## Maths and Magic on Radio 4...

posted by Kevin Houston

Should have posted this the other day: Last Friday Radio 4 transmitted a programme on maths and magic entitled, Maths and Magic! It features Jolyon Jenkins investigating the connection between the subjects in the title. You can hear it here for at least a few days. There are links on the web page but I thought I should add some. One of the magicians, Alex Stone, wrote one of my favourite books of the year, Fooling Houdini, which is about how he realized he was a terrible magician and devoted himself to changing that by immersing himself in the study of magic. He also does the same trick as Jenkins and there is a deeper explanation of it in the book. The Radio 4 website also lacks links to MathsJam and to James Grime, a contributor to Numberphile. Furthermore, the website cites Persi Diaconis as the sole author of Magical Mathematics when in fact it was co-authored by Ron Graham. And on the subject of Radio 4, today’s Infinite Monkey Cage isn’t about maths but is well worth a listen because of the excellent contributions from James...

## Singh, Simpsons and Skeptics...

posted by Kevin Houston

This is a rather belated review of Simon Singh’s talk on The Simpsons and Their Mathematical Secrets given for the Leeds Skeptics in the Pub in the Victoria Hotel pub. (There’s a chance to win the book, details below.) The Victoria is not a large pub so I did wonder where the audience was going to fit. In the end I counted nearly 100 people and this did not include those standing in the corridor or in the street outside. The Sardine Conjecture for @SLSingh at @leedsskeptics pic.twitter.com/XqE0p3lOs5 — Martin Iddon (@WalrusofComms) October 20, 2013 The talk is in two halves, the first covers the book and in the second Singh bravely invites questions on any topic connected to his work. The book part of the talk, accompanied by slides, delves into the mathematical backgrounds of the writers of The Simpsons and Futurama and how they inserted mathematical jokes into the programmes. Some of this was familiar as I’m a fan of both shows (though The Simpsons seems to have jumped the shark a while ago — Ned secretly married Mrs Krabappel!?) but Singh has done his homework and gained access to the writers so there is plenty here that was new to me. One of my favourite bits from the book wasn’t mentioned in the talk — there was a theorem invented purely for an episode of Futurama. So, the first part of the talk was about the book and during the interval Singh signed copies. Afterwards he did a question and answer session which amongst other things covered libel laws, Fermat’s last theorem, the Big Bang and quack medicine. The tour continues for another month, details can be found here. A great talk – you won’t be disappointed if you go. —-...

## Interview with Richard Elwes...

posted by Kevin Houston

In this post I interview Richard Elwes, author of a number of maths books in recent years. He has kindly agreed to give away signed copies his two most recent, Mathematics in 100 Key Breakthroughs and Chaotic Fishponds and Mirror Universes. For a chance to win all you have to do is sign up for my newsletter, see the right sidebar or the bottom of this post. I’ll get my daughter to randomly pick two winners from the list of subscribers in a couple of weeks. On with the interview… Hello Richard! I think we first met when you were a PhD student here at Leeds. Can you give a quick account of your background to help people get a feel for where you are coming from? I initially followed a conventional academic path: I studied maths as an undergraduate at Oxford, and then moved north to Leeds to start a PhD in 2001, where I met you among other excellent people. My research at that stage was in the general field of mathematical logic. Afterwards I held a postdoctoral position in Freiburg in Germany, and at some point after that I somehow found my way into writing about maths for the general public. Apart from books, I’ve written for the New Scientist, and for the excellent Plus magazine online. In fact I got my first break there, when I won their New Writers’ competition in 2006, with an article about the classification of finite simple groups (http://plus.maths.org/content/os/issue41/features/elwes/index). Your book Maths in 100 Key Breakthroughs came out last month. What’s it about? Who is it aimed at? It’s aimed at anyone who finds the idea of maths interesting and appealing, but who doesn’t want to drown in equations and jargon. So it’s written with...

## Simon Singh and the Simpsons...

posted by Kevin Houston

The week has been so busy that posting this has been delayed – I’ve been to London, Birmingham and Sheffield in the last three days. I did manage to install for a newsletter sign-up form for the blog. You can see it in the sidebar unless my hacks to the code have destroyed it for your browser. Please sign up if you want to know about maths articles, events, etc. Back to business. Simon Singh‘s new book, The Simpsons and Their Mathematical Secrets, is as the title suggests about the mathematics occurring in the long running cartoon The Simpsons. Now, I knew there were plenty of maths gags in the Simpsons but I’m surprised he has filled a whole book. Maybe he also includes stuff from Futurama which was packed with maths gags and even led to a lecture by Sarah Greenwald. This became a DVD extra on Bender’s Big Score (The audience features Simpsons and Futurama’s creator Matt Groening.) Singh’s book is not out yet but you can read an article about it in last Sunday’s Observer. He’ll be doing a (rather small) tour to promote the book, details are on his website. I’m planning to be at the Leeds event so maybe I’ll see you...

## How to Think Like a Mathematician

posted by Kevin Houston

It’s the blog’s birthday! It’s also the week before the start of the university year here and so it’s a good time to shamelessly advertise my best selling book. The success of How to Think Like a Mathematician has taken me by surprise. It has sold nearly 10,000 copies since it was published. Ok, I’m not going to rival JK Rowling but given that the statistic often quoted in the publishing world is that 95% of books sell less than 5000 copies, I am really happy that the book has found an audience and I get emails from all round the world saying how much they like the book. It has been translated into French, German, Turkish and a Korean version will appear soon. (See links below.) It has been adopted as a textbook throughout the world – I’ve lost count of the places that use it. Enough of the puff, what’s it about? Well, the book was written for anyone studying a mathematics degree or a mathematically based subject. My aim when I wrote it (it took me about 17 years but that’s another story) was to let students in on some of the secrets behind actually thinking like a mathematician so that learning mathematics becomes easier. The book has six parts: Study Skills for Mathematicians. This shows you how to read and write mathematics. The latter is an easy to learn skill and will set your work apart from most other students. How to Think logically. Mathematics is well known to be dependent on logic. Here you learn the basics from a very practical perspective. Definitions, theorems and proofs A key difference between university and pre-university level mathematics is that it is now not just about calculating. Definitions are made precise, mathematical truths (called theorems) are stated and they have to be shown to be unequivocally true by giving a proof. In this part you learn how to pull apart definitions, theorems and proofs so that you understand them. Techniques of proofs. Now that you have to prove things you need some techniques to bring out as the need arises. Mathematics that all good mathematicians need. The title is self-explanatory. Many other books introduce university level mathematics but then forget to include the maths you will be studying in your course. Closing remarks. We pull together everything learned so that rather than rely on superficial rote learning you develop deep understanding. Some of the tips in the book are simple but effective. The following is a taster: Many students asked to show that an equation holds will merely rearrange it an haphazard way to produce some other equation they know is true such as 1=1. (The reason why this is just plain wrong is given in Chapter 21, Some Common Mistakes. Hint: it is to do with logic.) Here’s a simple tip that works so much better: Take the most complicated side and do something to reduce it to the other side. This works. It stops you tying yourself in knots when randomly rearranging terms from one side to the other in the hope that something will happen. Instead it forces you to think about what you can do with the terms. It’s a simple trick that changes behaviour. If you are interested in owning a copy, then just follow the links below: How to Think Like a Mathematician: A Companion to Undergraduate Mathematics (UK) How to Think Like a Mathematician: A Companion to Undergraduate Mathematics (US) Wie man mathematisch denkt: Eine Einführung in die mathematische Arbeitstechnik für Studienanfänger (Deutschland) Comment penser comme un mathématicien (France) Matematikçi Gibi Düşünmek (Türkiye) How to Think Like a Mathematician: A Companion to Undergraduate Mathematics (Italia) How to Think Like a Mathematician Paperback: A Companion to Undergraduate Mathematics (España) (Some of the above are affiliate...

## Maths predicts – this time movie success

posted by Kevin Houston

In a previous post I talked about predictions using maths and Nates Silver‘s book on essentially that topic was one of my favourites books of last year. This next one is a bit of fun – predicting movie success. Film buffs will know William Goldman’s quote about making films “Nobody knows anything” which is taken to mean that no one can predict how a film will do at the box office. However, researchers have some good news. Mestyán, Yasseri and Kertész have published Early Prediction of Movie Box Office Success Based on Wikipedia Activity Big Data. As you can tell from the title the key is to use online data and activity. Their algorithm gives good predictive power up to a month before the film is released and hence will be of little use to Hollywood producers receiving pitches for new films. Of course, these are early days and even the Oxford Internet Institute news article uses the word predicts in quotes. Nonetheless the authors compare their results to those obtained using Twitter by other researchers and find it better. The paper is freely available and since the maths behind it is accessible to undergraduates it would be great for a student mini-project. (Talking of projects, over the summer I had an undergraduate studying symmetry matching and it has turned out very well so I’ll definitely be writing about that soon.) Photo attribution: Alex Eylar,...

## Glassified – Reinventing the ruler?...

posted by Kevin Houston

The ruler has remained unchanged for thousands of years. Now it has been updated! You can read more (but not much) at Anirudh Sharma’s website, Creative Applications, Gizmag, and...

## Maths Jam plug and Zeno’s Paradox (not together!)...

posted by Kevin Houston

First a plug for the next Maths Jam in Leeds: It will be on Tuesday 23rd at 7pm in the White Swan (it’s pretty much next to the City Varieties). I’ll be there (I know that if don’t publicly commit to it, then I’ll miss it) so maybe I’ll see you there! Here’s a link to a rather good Ted Ed video on Zeno’s paradoxes. I wish I had seen this before I gave my History of Maths course this year, it would have been quite useful despite its simple nature. There is a page for further information on the this...

## Sconic sections

posted by Kevin Houston

This is a strange one: How to make edible conic sections! You can find the details at Evil Mad Scientist. If you enjoy maths-related cooking, then see my post on Tau-nados. Does anyone else have any other...

## Wolfram on Leibniz

posted by Kevin Houston

The usual busyness at the end of the teaching term and a holiday last week have meant that I’ve not been posting for a while. Today’s post is a short one from me directing you to a fairly long post by Stephen Wolfram on his recent visit to the Leibniz archive in Hannover. Read it...

## Lagrange book sale

posted by Kevin Houston

I never miss a chance to rummage around in second-hand book shops. In the past bargains were easy to come by but now the internet has killed that off. Now all books are priced pretty much the same and I no longer have the experience of approaching the counter carrying a much-underpriced book with the feeling that I am stealing from the shop and am about to be discovered. Those days are gone. As they are not experts, unfortunately charity shops price their books by consulting the web. This leads to setting the price of some dog-eared copy at just below the price of a mint condition one. Also, I miss the end-of-search feeling as I come across a long sought-after book. These days if I want a book I can find it on Amazon or Abebooks in minutes. The latter is my favourite second-hand book-seller site. They often send me emails about books and a recent one is worth sharing. One of the most expensive sales on the site in March was a book by Lagrange. The relevant part of the article is the following. Our list also includes an historic textbook from 1788 that has had a lasting influence on mathematics. Sounds a bit dull? Not at all. Méchanique Analitique by Joseph Louis Lagrange sold for $13,112. Born in Italy, Lagrange became a famous academic in France and Germany, and managed to survive the French Revolution despite the carnage surrounding him. Méchanique Analitique advanced analytical mechanics beyond the work of Isaac Newton and Galileo Galilei. He wrote the book while in Berlin where he was director of mathematics at the Prussian Academy of Sciences. It was his greatest piece of work, although he contributed widely to mathematics and astronomy. He laid down...

## Colin Wright and the mathematics of juggling...

posted by Kevin Houston

Recently, an acquaintance from my days as a researcher at Liverpool University alerted me to the existence of the Museum of Mathematics in New York. My acquaintance, Janet West, was a PhD student when I was at Liverpool and is now involved in the museum. There’s plenty of stuff online to look at but I would like to draw your attention if you have not already seen it to a lecture by Colin Wright. Colin is well-known in the mathematics communication community as he probably does more mathematics talks in schools around the country than anyone else. His main talk is about the mathematics of juggling. You can see him talking on to the BBC about it by clicking this link. (Note that the headline says he is a teacher whereas in fact his job is in marine navigation!) Colin gave a talk at the Museum of Maths in New York which was recorded and is on YouTube. You can even buy a DVD version. Teachers: If you are interested in seeing his talk at your school, then go to his...

## What is the Best Proof of Cauchy’s Integral Theorem?...

posted by Kevin Houston

Today’s post may look as though I’m going all Terry Tao on you with a long post with lots of mathematical symbols. It’s really about the learning and teaching of Cauchy’s integral theorem from undergraduate complex analysis, so isn’t for everyone. If it’s not your cup of tea/coffee, then pop over here for some entertainment. Cauchy’s Integral Theorem Cauchy’s Integral Theorem is one of the greatest theorems in mathematics. There are many ways of stating it. Here’s just one: Cauchy’s Integral Theorem: Let be a domain, and be a differentiable complex function. Let be a closed contour such that and its interior points are in . Then, . Here, contour means a piecewise smooth map . In my years lecturing Complex Analysis I have been searching for a good version and proof of the theorem. My definition of good is that the statement and proof should be short, clear and as applicable as possible so that I can maintain rigour when proving Cauchy’s Integral Formula and the major applications of complex analysis such as evaluating definite integrals. Many of the proofs in the literature are rather complicated and so time is lost in lectures proving lemmas that that are never needed again. Here’s a version which I think has a good balance between simplicity and applicability. I’ve highlighted the difference with the version above. Cauchy’s Integral Theorem (Simple version): Let be a domain, and be a differentiable complex function. Let be a simple closed contour made of a finite number of lines and arcs such that and its interior points are in . Then, . Here an important point is that the curve is simple, i.e., is injective except at the start and end points. This means that we have a Jordan curve and so the curve has well-defined interior and exterior and both are connected sets. With this version I believe one can prove all the major theorems in an introductory course. I would be interested to hear from anyone who knows a simpler proof or has some thoughts on this one. Proof of Simple Version of Cauchy’s Integral Theorem Let denote the interior of , i.e., points with non-zero winding number and for any contour let denote its image. First we need a lemma. Lemma Let be a simple closed contour made of a finite number of lines and arcs in the domain with . Let be a square in bounding and be analytic. Then for any there exists a subdivision of into a grid of squares so that for each square in the grid with there exists a such that Proof of Lemma The set up looks like the following. For a contradiction we will assume the statement is false. Let and divide into 4 equal-sized squares. At least one of these squares will not satisfy the required condition in the lemma. Let be such a square. Repeat the process to produce an infinite sequence of squares with .By the Nested Squares Lemma (which is just a generalization of the Nested Interval Theorem) there exists . As is differentiable there exists such that there exists a grid of squares covering . Let be the set of squares such that and let be the set of distinguished points in the lemma. Define by Then as is differentiable, is continuous (and hence integrable). Without loss of generality we can assume that is positively oriented. Let be the union of positively oriented contours giving the boundary of . Since is made of a finite number of lines and arcs will itself be the union of a finite number of lines and arcs. For such that , is just the boundary of a square. On we have As is the derivative of by the Fundamental Theorem of Calculus and the fact that is closed we get Now, and edges of touching squares will cancel. So by (1) and (2). We...

## Mary Cartwright article...

posted by Kevin Houston

Mary Cartwright is fairly well-known amongst mathematicians in the UK but less widely known amongst the general public. A recent BBC online article about her and her work may be the beginning of a change in this situation. There is even a Radio 4...