The months are drawing in

February was two days shorter than January. "I'm worried", I tweeted, "If this carries on, how long will December 2012 be?"

Another way of looking at this is that February is about 93.5% the length of January, so I asked which would produce a shorter December:
A. losing a fixed 2 days each month; or,
B. each month being 93.5% of the previous.

It's possible, of course, to simply calculate the answer. However, it is possible to come to an answer as to which is shorter without recourse to such a messy technique.

Under B, we know 93.5% of January is two days, the amount by which February is shorter. If March is 93.5% of February we know this decrease must be less than two days because February was shorter than January. And so on. The decrease in A is always 2 days, but the decrease in B is 2 days in the first month and less for later months. Since the overall decrease has been greater, A gives a shorter December.

I suppose there's a niggle that we don't usually allow fractions of days on the calendar, so if you're going to be all 'real world' about it then each month should be rounded and this rounding will occur before the 93.5% is calculated to form the next month. So I suppose we will have to do a messy calculation after all.

Under A, losing 2 days per month for 11 months is 22 days, so December will be 9 days long.

Under B, taking each month to be 93.5% of the previous, and then rounding to the nearest integer in the normal way, I get a sequence for the number of days for each month: 31, 29, 27, 25, 23, 22, 21, 20, 19, 18, 17, 16.

So my December is a full seven days shorter by the 'fixed two days' method.

Did you get 14 or 15 days for December? If you simply take each month to be 93.5% of the previous without rounding, you calculate 0.935^11*31 and get December as 14.8 days. You can round this to 15, or take the whole days to get 14, but this requires nonsensical things like 18.5 days in November to have happened on the way.

I feel as though this could be a nice, silly way in at various levels to either some basic arithmetic, exponential decrease, through to boundedness in geometric sequences. Even, into some discussion about translating mathematics to real world answers, as the quick 0.935^11*31 calculation masks a whole mess of unreality along the way.

London Day Trip Stop 1: British Library

In a previous blog post Things to do in London on a Tuesday I asked for suggestions of things to do on my day trip to London. I went because I was invited to attend the inaugural London walking tour from Maths in the City - we'll get to that - and apparently the date was chosen on purpose as it was a palindrome: 21-02-2012.

On arrival in London, I idly tweeted a photo along the station platform with the caption "Looks like London". James Clare responded to this with a guess at which station.

I was at St. Pancras. What everyone seems to notice about St. Pancras is the roof, seen in the picture below which I took on the day. This was apparently first opened in 1868 and the 243ft roof created "the largest indoor space in the world". More recently, an £800m restoration project was completed in 2008. You can watch an interesting BBC short video about this project, featuring interviews with the chief architect, the project director and the project engineer.

Following James' tweet, I liked the idea of a guessing game so I tweeted a clue for my new location. David Ault guessed correctly that I was at the British Library.

Outside the British Library is a statue of Newton (1995), which had been suggested to me as a destination on my day trip. Designed by Sir Eduardo Paolozzi, who said it was "intended to show how art and science are interconnected", the statue is inspired by a 1795/circa 1805 colour print finished in ink and watercolour on paper entitled "Isaac Newton" by William Blake, which can be found in the Tate gallery.

Inside the library, I found the King's library. Created by George III, donated to the nation in 1823 by his son George IV and once housed in the British Museum, these books are housed in an eye-catching six-storey tower (pictured below). A description of how the library was formed and its history is available on the British Library page George III Collection: the King's Library.

I also visited the Treasures of the British Library gallery, described on the website as "a permanent free display of many of our greatest treasures". No photographs were allowed but I took a few notes.

I saw a collection of photos and documents from the Scott polar expedition. Fresh from my Twitter photo clue competition, a note caught my eye about the use of photography to increase public interest in Scott's expedition. The Guardian has a piece about an exhibition of photos from the expedition, which says:

In 1910 and 1911, as Scott struggled to raise funds and public support for the Terra Nova venture – media hysteria about the race to the pole was the reason the South Pole was bolted onto the scientific expedition – the explorer knew the propaganda value of superb images

Herbert Ponting (1870-1935) was hired as expedition photographer. A selection of Ponting's photos have been uploaded to a gallery by the National Archive and one is available below.

I also saw two pages of notes by Leonardo Da Vinci from Codex Arundel. Leonardo began the collection in 1508, writing that this was "a collection without order, drawn from many papers". The writing is mirrored Italian written from left to write. According to the British Library website pages were added from different periods in Leonardo's life, "covering practically the whole of his career". The website has this to say of the contents:

It includes notes for a book on the physical properties and geographical effects of water, and a broad range of other material encompassing Leonardo’s other interests in art, science and technology over a period of four decades, from the description of a prehistoric sea monster (c. 1478-80) to architectural projects for the royal residence at Romarantin in France (dating to about 1517/1518). The range of subjects - from mechanics to the flight of birds - demonstrates Leonardo's almost compulsive intellectual curiousity about scientific and technical matters.

The pages I saw in the library were on mechanics and arithmetic. There are pages on the British Library website that show some pages from the Codex Arundel, and an introduction to the Codex.

I also saw an exhibition on early printing, many sacred texts and Magna Carta before moving onto my next stop. That, I'll save for another post.

Stereotype-abiding mathematicians of the world, unite!

Recently I wrote a post, Mathematicians are people too, about the image problem of mathematicians and called for examples of mathematicians who do not fit the traditional stereotype.

On Google+, Christian Perfect said:

ok, so, as an autistic white male mathematician, I'm going to steer clear.

I said that as a glasses-wearing, bearded white man, I didn't feel much use either. Christian replied:

so: stereotype-abiding mathematicians band together to reassure public that mathematicians don't necessarily conform to the stereotype.
That's the kind of logic only mathematicians would appreciate.

I also received this comment from Twitter user @sebmr2:

Didn't Galois do enough to break stereotypes for me to fit them? 

I don't think all mathematicians should personally break the stereotype. I remember some years ago I was working in a university mathematics department and someone had pinned up a newspaper comment piece in the staff room about how lecturers should dress in sharp suits like businessmen if they want to give the right impression to their students. I don't agree with this.

However, my call for examples was written from another viewpoint. Not: can I, as someone who studied mathematics at university, adapt myself to avoid the stereotype. Instead: what if I was faced with a class of students, many of whom would never fit the stereotype (by virtue of their ethnicity or gender, for example)? I would want my class to believe that they too could be mathematicians, yet if they think all mathematicians conform to a certain 'type' then this is a barrier to them seeing themselves in this way. Particularly as it is obviously an incorrect stereotype.

So I am interested in breaking stereotypes not to change you, dear reader, but to better inspire others.

To finish, I would like to share a video suggested on Google+ by David Roberts. The video of Nalini Joshi is by Trixie Barretto, who says of it:

There's a mathematician six floors above me where I work. I'd never had much to do with her, but I'd heard she'd had an unusual childhood in Burma, and grew up to become the first female professor of mathematics at the university where we both work. One day on Twitter she wrote, "Maths is in my heart," a sentiment both alien and amusing to me, being someone who's terrible with numbers. It stayed with me though, and later that afternoon I knocked on her door and asked if she'd tell me her story.

Picture this!, an interactive problem/puzzle

"PSUM: problem-solving in undergraduate mathematics", a project we are supporting at work, have produced an interactive problem/puzzle as an early prototype for a virtual problem solving resource package.

Picture This! is an interactive problem solving application. You are shown a diagram which somehow relates to two integers. You are asked to change the two integers and explore the effect on the diagram. Once you have figured out the effect of the two numbers on the diagram, you are invited to consider a series of probing questions, such as:

What is allowed to change and what must stay the same? Do different pairs of integers necessarily lead to different diagrams? Can you start from a diagram and work back to the initial numbers? 

Important: Once you have played with the virtual problem solving environment, please fill out this survey from the researchers. The researchers have said to me that they are happy for the page to be public and hope that anyone who uses it will fill out the survey. Doing so, you will help the researchers discover whether the use of this software to present problems is worthwhile and beneficial. The survey asks if you are a student or a tutor. If you choose "student" you will be asked about your use of the simulation and your understanding of the underlying mathematics. If you choose "tutor" (or leave the question blank) you will be asked about how you used it with undergraduate students.

This project seeks to produce "a virtual problem solving environment which hosts problems suitable for a range of undergraduate mathematics courses". If you want to find out more about this project then you can read the interim report from this project over on my work blog.

Almost all above average

This morning James Grime tweeted about a BBC News article, "'Third of UK postcodes' have slow broadband speeds". This quotes Julia Stent, director of telecoms at uSwitch saying:

Britain might be riding the wave of a super-fast broadband revolution, but for 49% who get less than the national average broadband speed, the wave isn't causing so much a splash as a ripple.

Now, the thrust of the article, that broadband speeds are undesirably slow in some parts of the country, might be valid, but the appeal to the "average" is a very weak argument (Update [23:47]: Although, please see the comment below). The result that 49% are below average should not come as a big surprise!

James, rightly, questions which average is most appropriate, but I am more interested in a tweet by Ian Preston:

We can make almost everyone above average if we are happy for one person to be really badly off.

This, of course, is quite right.

Unless I'm reading it wrong (and I may well be), the Bank of England's Lending to Individuals December 2011 has outstanding net lending to individuals as £1451.4 billion. The UK Office for National Statistics gives the Public Sector Net Debt excluding financial interventions as £988.7 billion (January 2012) and Total population (UK) as 62.3 million (mid-2010).

If we gave one person all that debt and everyone else zero, then a simple average would be £2.44 trillion divided by 62.3 million people, which is £39,167. 

Since almost everyone is worth zero, we would almost all be 39 thousand pounds above average. Sound good? This would sort out Government debt and make almost all of us "above average". And if we're "above average" then, erm, everything is fine, right?

(Flaws in the argument left as an exercise for the reader!)

Barriers to teaching

Lecturer in Mathematics.
School of Mathematics, University of Excellence.
Competitive salary.

Applications are invited for the post of Lecturer in Mathematics.

The University of Excellence is ambitious for the future, priding itself on its commitment to world-leading research and investment in an outstanding research environment. The opportunity is available to join a dynamic, highly esteemed and international research programme. Candidates who can interact with one or more of the School's existing research strengths are particularly encouraged to apply.

As a successful candidate, you will have a PhD (or equivalent) in some branch of Mathematics and a track record of relevant research. You will have demonstrated the ability to publish consistently in leading research journals and be able to provide evidence of your experience attracting research funding. You will advance the School's research agenda by supervising a group of PhD researchers.

In the latest UK Research Assessment Exercise the School submitted research output from over 70 staff. 65% of this research was recognised as being either world-leading or internationally excellent in terms of originality, significance and rigour.

Underpinned by the quality of its research, the School offers a range of degrees from undergraduate to postgraduate level. The successful candidate will also be expected to contribute to the development and delivery of teaching in Mathematics.

Potential candidates are encouraged to check our website for full details.

Reading around the Alan Turing Pardon

I have a piece in this week's Pod Delusion episode 123 at 45:00 on the pardon for Alan Turing.

Here are links to some of the bits I talked about in this.

I spoke about concerns of overdoing the Turing celebrations, saying: what Turing did was brilliant, but we should celebrate what Turing actually did, not some imagined feats, and we should not forget others in doing so. You can read more about this and find out about the article which suggested that had Turing lived then Silicon Valley might have been started in the UK at 'Beware the Alan Turing fetish' by John Graham-Cumming.

Turing was convicted under Section 11 of the Criminal Law Amendment Act 1885. In 2009 Gordon Brown issued an official apology for the way Turing was treated. Read about the official Government apology in 'PM's apology to codebreaker Alan Turing: we were inhumane'. Read how the apology came about in 'How Alan Turing Finally Got a Posthumous Apology' by John Graham-Cumming.

Now there is a current e-petition calling for a pardon for Turing. John Leech MP issued an early day motion calling for this pardon. (I also mentioned the current e-petition calling for a pardon for Oscar Wilde.)

Asked a question in House of Lords, a Government Justice Minister said "a posthumous pardon was not considered appropriate". Read the text of Lord McNally's statement.

I've seen the refusal to pardon Turing described as "homophobic" and an "act of malice". Particularly, the complaint is that Turing is still seen as a criminal in the eyes of the law.

John Graham-Cumming on 'Why I'm not supporting the campaign for a pardon for Alan Turing', in which he writes about the Protection of Freedoms Bill, which "specifically allows for the disregarding of convictions under the old law that was used against Turing".

To honour Turing I suggested you might attend events under the Alan Turing Year banner, or donate to Bletchley Park's Action This Day! fundraising campaign.

This piece used audio from episodes 84 and 85 of the Pulse-Project Math/Maths Podcast.

A puzzle from James Grime about abcdef

Today James Grime tweeted this question/puzzle:

Is there a six digit number abcdef such that the following all hold?

1. a+b+c+d+e+f = y
2. ab+cd+ef=10y
3. abc+def=100y

If not, show why not.

A little tweeting back and forth verified that "ab" means 10a+b not a×b.

If you want to have a go at this, don't read any further until you have!

---

First, rewrite the expressions so that both sides use standard arithmetic:

1. a+b+c+d+e+f = y
2. 10a+b+10c+d+10e+f=10y
3. 100a+10b+c+100d+10e+f=100y

I noticed that since a, b, c, d, e and f are all positive integers, so must y be. Then 10y must end in 0 and 100y must end in 00.

From 3, we see that in 100y only c and f contribute to the units, so c+f=0, or f=-c. See also that only b and e contribute to the 10s, so b+e=, or e=-b. 
Since a, b, c, d, e and f are positive digits 0-9, the only values that satisfy these equations are c=f=0 and b=e=0.

From 2, we see that in 10y only b, d and f contribute to the units. Therefore b+d+f=0, or d=b+f. Since b=f=0, we know that d=0.

So solutions to this problem have a being any digit 0-9, b=c=d=e=f=0, and so y=a.

The answer isn't no, nor is it quite yes. These multiples of 100000 are not exactly interesting!

---

After I emailed Jim I was interested to see this solution, posted at The Wandering Monster, which took a substitution approach. On Twitter Dave Hughes gave his approach: "My solution was inelegant - I threw a C# program at it". How interesting to see how different people approach a problem!

Update (20 mins after posting!): Please check the comments for a caveat I've missed.

Things to do in London on a Tuesday

Next Tuesday I will spend a day off in London. I am asking people to offer suggestions for things I could do with my time by adding pins to this Google Map: PR's Day Off. A few people have already added their suggestions but it would be great to hear more.

View PR's Day Off in a larger map

Load the map, then you need to log in with your Google Account then a big red edit button should appear and you can add a pin. The description I put on the map:

Ground rules:
Date is Tuesday 21st Feb.
Free or low cost. It's expensive just to get to London.
I have a zone 1 & 2 tube card.
Train times are non-negotiable (fixed tickets).
I am going on the Maths in the City tour.
All else is up for grabs.
Make suggestions for places to go and things to see by putting pointers on the map. Explain why and give timings and a link if appropriate.
For timed events: Remember to allow enough time for travel (for someone who doesn't know where they are going).
Please don't delete other people's entries. If you disagree, by all means leave a comment to say so but don't delete something you haven't put there.

If you can't get the Google Map to work please leave your suggestion in the comments of this blog post.

George and Julian

Yesterday, the @mathshistory Twitter feed tells me, was the anniversary of the birth of Julian Schwinger (1918-1994), one of the great physicists of the 20th century. (Technically I queued this tweet up but there are a lot of days and a lot of mathematicians to remember...)

Schwinger is known to me particularly through his connection to the story of George Green. Green was a Nottingham mathematician who did work on electricity and magnetism (among other things) that, largely unrecognised in his lifetime, was discovered and brought after his death to further attention by William Thomson (later Lord Kelvin). The application of Green's work in 19th century science was impressive but it found a new legacy in the 20th century.

At the 1993 celebration in Nottingham of the bicentenary of Green's birth, Schwinger spoke about his use of Green's work (a talk written up as The Greening of Quantum Field Theory: George and I).

Schwinger's account is worth reading. He describes his use of Green's work first on microwave radar during World War II, then in the development of the microtron and synchrotron particle accelerators, and finally to solve a problem on quantum electrodynamics, work which earned him a share, with Sin-Itiro Tomonaga and Richard Feynman, of the 1965 Nobel Prize for Physics.

In the preface to his most famous work, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (1828), Green had written:

Should the present Essay tend in any way to facilitate the application of analysis to one of the most interesting of the physical sciences, the author will deem himself amply repaid for any labour he may have bestowed upon it.

Schwinger's account helps us to understand how Green not only impacted the physics of his age, but how it continued to have impact beyond anything Green could have imagined.

Mathematicians are people too

Last Saturday in the Telegraph there was a feature announcing the start of a numeracy campaign: Make Britain Count. This included an article by Rachel Riley about "the stigma around maths". She writes about the "image problem" of maths and numeracy:

I’m a blonde Essex girl, so I’m well used to being talked down to, but when I tell people I did a degree in mathematics at Oriel College, Oxford, I see their jaws hitting the floor. Mathematicians labour under a negative stereotype – older men in anoraks with beards and glasses. Maths isn’t sexy. 

She talks about problems of attitude and relevance to the real world, and the need for creative teaching to

teach children number skills from first principles. They have to know the underlying “why” of maths, not just memorise the formulas.

Let's talk a little about the issue of the image of mathematicians. Last night on Twitter I was approached by user @philhumpo, a teacher from Exeter, with this query: "I need a 'top 5 crazy mathematicians' (duelling Romans, drowning kittens etc)."

This sort of thing concerns me. I wondered in what sense he meant "crazy". Mathematics can seem to have an association with mental illness in popular culture and so I'm naturally concerned if "crazy" is being handled sensitively. Also, many of the interesting historical anecdotes turn out to be false or exaggerated, an issue touched on in my previous post.

Thankfully, it was just an issue of the brevity of messages on Twitter. Phil explained the heart of the problem. It's the last day of term today and Phil has his class of 15 year olds for a shortened lesson. He has discovered many of them think "all mathematicians are grey suit baldies with social problems" and hopes to disabuse them of this view.

With the reference to duelling mathematicians, Phil is clearly aware of Évariste Galois, who clearly has a romantic and stereotype-breaking story. Ramanujan is another good story. You can find online biographies of women mathematicians - Ada Lovelace, Mary Somerville and Sophie Germain are typical examples, though there are many more.

I also wondered about more contemporary sources. Recently I came across a photo blog "This is what a scientist looks like" via the @HESTEM Twitter feed. A quick search reveals just one mathematician featured so far. As Phil put it "hmmm… not a duelling Frenchman but not a grey suited baldy that's for sure".

I recommended Katie Steckles' video Playing Games with Squares. Katie certainly doesn't fit the stereotype and the video shows her having fun with mathematics.

There are a host of careers profiles from a range of different people in the Maths Careers Career profiles, where just scrolling down the page gives an idea of some of the stereotype-breaking people involved with mathematics, and a similar list is available with the Plus Careers Interviews.

I am sure there are countless more examples of mathematicians breaking the mold - mathematicians really are people too! - and I've only had a quick think about it. Perhaps you can suggest your favourites in the comments.

Why do we enjoy maths history misconceptions?

I don't think I have come to a conclusion from my previous blog post about historical accuracy and popularisation, though there were some interesting points in the comments (relating less to my comments about the 'errors which may be gently corrected' and more to the 'demands of the narrative').

George Jelliss and Thony C. both read the famously inaccurate Men of Mathematics by E.T. Bell in their youth and were inspired to mathematical lives as a result.

Will Daniels suggests I should hold different standards for different people, so those writing historical research are held to a higher level of accuracy than those writing for a popular audience. I'm not sure this feels right. Thony asks a really interesting question:

is it possible to achieve the inspiration generated by Bell's book and be historically accurate at the same time?

I think this is at the heart of the matter. If it is possible to inspire through popularisation while remaining completely accurate then I can safely hold everyone to this high standard. However, if inspiration requires a little showmanship, if telling a good tale means not getting lost in minor distractions and sub-clauses, then we have our double standard.

This brings me to a final, anonymous comment that includes the following statement:

I am a great believer in the wisdom of stories, regardless of their provenance. If some stories persist despite being disproven, there must be a reason.

My first reaction on reading this is that it is preposterous. If you start presenting stories you know to be disproven you are in the realm of historical fiction. Historical fiction is fine, but these are now just stories and have no place being presented as real accounts of historical mathematics and mathematicians. Then today I was struck by something relevant.

I was listening to Paul Dirac and the religion of mathematical beauty on the Royal Society Library podcast while wrangling with the washing machine. This recording, of a talk given in March 2011 by Graham Farmelo, covers the life of Paul Dirac. Farmelo talks about how Paul Dirac is considered to be the theoreticians' theoretical physicist, yet he had a very practical schooling and took a practically-focused engineering degree. Farmelo says (15:40):

Let's get one thing right, he was a very practically-minded person. Completely different from the image that he has among most theoretical physicists.

This is as so often the case; the established fact isn't just slightly wrong but completely wrong. This is the case in the story that Einstein did poorly at school, a misconception that Thony C. tells me is not as well known as I thought it was when I used it as an example in my previous post.

Do we really want to believe Dirac is a theoretician with no practical sense, that Einstein was a terrible student made good? Is there really some "wisdom" in these stories that causes them to "persist despite being disproven"?

Rather than necessarily being wise, I think we are drawn to certain types of story. The Dirac perception reinforces the view of a flawed genius; a theoretical physicist with no sense of the real world. The Einstein story perhaps speaks to a desire for the plucky underdog to win out in the end.

Aren't these classic Hollywood ideas? Do other common misconceptions fit into the Hollywood-style? (Galois' heroic struggle against the odds to invent Galois theory in a single night before the dual springs to mind. What others?) Do, in fact, stories that deviate from historical record and persist deviate when the story fails to fit a certain sort of narrative?

Perhaps more importantly, are there correct historical stories which fit a classic Hollywood narrative? (I'm thinking, for example, of George Green teaching himself advanced mathematics "in the hours stolen from [his] sleep".) Perhaps stories of this type are the key to achieving Bell-like inspiration while maintaining historical accuracy.