Plus Blog

September 5, 2014
Double pendulum

Image by Michael G. Devereux.

This week's striking image is a long exposure photograph of a double pendulum by Michael G. Devereux. Double pendulums provide a striking physical illustration of chaotic behaviour, captured in this image by attaching LEDs to each part of the pendulum. You can read more about chaotic behaviour in this and other systems in Finding order in chaos.

The picture is one of the images that appear in the book 50 visions of mathematics, which celebrates the 50th anniversary of the Institute of Mathematics and its Applications.

You can see previous images of the week here.

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September 4, 2014

Adding fractions is probably the first difficult bit of maths we come across at school. For example, to work out $\frac{5}{6} + \frac{7}{10}$ you first need to figure out that the lowest common multiple of 6 and 10 is 30, and that in order to get 30 in the denominator of both fractions you need to multiply the numerator 5 by 5 and the numerator 7 by 3. This gives

$\frac{5}{6} + \frac{7}{10} = \frac{(5 \times 5)}{30} + \frac{(7 \times 3)}{30} = \frac{25}{30} + \frac{21}{30} = \frac{(25+21)}{30} = \frac{46}{30}.$

You then need to get rid of the common factors of 46 and 30, giving the final result $\frac{23}{15},$ which bears no resemblance whatsoever to the original two fractions. Doing this as a ten-year-old who has never seen it before is pretty tough.

Here is an alternative recipe that always works and doesn't involve faffing around with lowest common denominators. Writing "top" for numerator and "bottom" for denominator, the idea is to do:

(top left x bottom right + top right x bottom left) / (bottom left x bottom right).

Applied to our example this gives:

  \[  \frac{(5 \times 10 + 7 \times 6)}{(6 \times 10)} = \frac{(50 + 42)}{60} = \frac{92}{60} = \frac{23}{15}.  \]    

The difference to the standard way of adding fractions is that you are not bothered with finding the lowest common denominator. You simply use the product of the two denominators as a common denominator. Then, in order to bring both fractions on that common denominator you only need to multiply the numerator of each by the denominator of the other. Easy!

Apparently this is how Vedic mathematicians in ancient India added up fractions. If you happen to speak German, you can also explore this method in musical form in this maths rap by DorFuchs. And even if you don't speak German, it's cute!

August 29, 2014

Image by Ole Myrtroeen and Gary Hunt.

This vortex looks so friendly, we made it our image of the week! It was formed in a physical experiment in which a saline solution (stained blue) was ejected upwards into a still fresh-water environment. The image was produced by Ole Myrtroeen and Gary Hunt, University of Cambridge, and appeared on the cover of vol. 657 of Journal of Fluid Mechanics, © Cambridge University Press.

The picture is one of the images that appear in the book 50 visions of mathematics, which celebrates the 50th anniversary of the Institute of Mathematics and its Applications.

You can see previous images of the week here.

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August 27, 2014
Science festival

The British Science Festival is kicking off in Birmingham on the 6th of September, and as usual there is some great maths in the programme. You can see all the maths related events on this flyer or find out more on the festival website.

From intelligent machines and animal emotions to phones that can detect illness — find out how mathematics shapes our world!

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August 20, 2014
secret garden

As we attempt to close our overstuffed suitcases for the flight home, we are very pleased that Plus too is overflowing with goodies from this years International Congress of Mathematicians in Seoul.

From the excitement of the announcement of the Fields medallists to meeting mathematicians from around the world, we've had a brilliant time. We've still got a few articles and podcast yet to publish, but for now, you can find all our coverage from ICM 2014 here.

Thank you to all mathematicians for their talks and to those who generously gave their time for interviews and explanations. A huge thank you to the local organisers of the conference, the staff at the media office and the many enthusiastic and helpful volunteers who made the event such a success. Thank you!

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August 20, 2014

"Ingenuity", "far reaching vision", "unerring sense", "deep curiosity" — and best of all, "extraordinary creativity". These are some of the words that have been used to praise this year's Fields medallists at the International Congress of Mathematicians (ICM). These words aren't specific to maths. They could be used to describe anyone whose work is about discovery and beauty; writers, poets, or musicians for example. If there has been one overarching theme at this ICM, it's just how creative a subject mathematics is. To imagine what has never been imagined before, spot new connections, look at things in a new light and find hidden patterns lies at the heart of doing maths.

A Rubik's cube

A great example is Fields medallist Manjul Bhargava, who in a flash of inspiration discovered that a centuries' old maths problem could be solved using a Rubik's cube, reducing long and tedious calculations, quite literally, to child's play. But singling out specific examples gives the wrong impression: discovering the undiscovered and creating new ideas is central to the work of every mathematician, whether they are presenting their work here at the ICM or working away at home.

What guides these creative minds is often a sense of beauty. "There is an aesthetic side," says Phillip Griffiths, winner of this year's Chern Medal. "You find out what the most harmonious properties of a [mathematical] structure are, and then you let those guide you." You might be wrong, of course, things might be more complex, or perhaps even simpler, than you think, but still, it's often aesthetic considerations that lead the way.

The creative aspect of maths is one of mathematics' best kept secrets. If you didn't have an inspiring teacher at school, then you probably think that maths is all about repetition with no room for imagination and exploration. "Many teachers still teach that way; here is the problem, here is how you solve the problem," says Martin Grötschel, Secretary of the International Mathematical Union. "So what you do is learn recipes, how to solve a quadratic equation, etc. But you are not taught why and how you get to this equation. The really successful countries, where maths education has improved a lot, have taken the approach of teaching students to figure out for themselves why things go this way or the other way." (You can listen to our interview with Grötschel here.)

Fields medallist Bhargava suggested something similar to us: perhaps maths students, rather than learning how to apply "finished" mathematical tools, should be allowed to play with maths problems that have inspired past masters, to develop their mathematical intuition and get a sense of the excitement and adventure of doing mathematics (see our interview with Bhargava to find out more) — the sense of discovering, and shaping, something new.

A lot of research is currently going into finding out how to best teach maths at school, part of it supported by the International Mathematical Union. Luckily we know lots of teachers who are passionate about creativity in maths. And we are currently involved in a project that aims to foster this creativity, an effort that has resulted in the website Wild Maths. Equally luckily, we know lots of mathematicians who are passionate about communicating the essence of maths to a wider audience (Bhargava is one of them). So perhaps it won't be long until mathematics' best kept — and most exciting — secret becomes public knowledge.

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