## Plus Blog

September 4, 2014

Adding fractions is probably the first difficult bit of maths we come across at school. For example, to work out 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

You then need to get rid of the common factors of 46 and 30, giving the final result 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:

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.

August 27, 2014

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!

August 20, 2014

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!

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 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.

August 13, 2014

One of the reasons we love maths is that at its core it doesn't care about who you are, where you are from, the language you speak, the colour of your skin or your gender. Mathematics is a universal language. We feel this as people doing mathematics. But as we attend conferences, read journals and interview the many wonderful mathematicians we meet, we are reminded that as women we are in a minority.

This is not to say, however, that there are not brilliant women mathematicians. Quite the contrary. We have been lucky enough to with and reported on the work of many – Corinna Ulcigrai, Julia Gog, Helen Mason and Nathalie Vriend to name just a few.

Maryam Mirzkhani, the first female Fields medallist

But currently in the UK only 6% of maths professors are women, despite women making up over 44% of maths undergraduates (see this paper from the London Mathematical Society). The reasons for this are not well understood but the problem seems to be exacerbated by the small number of women itself: fewer women in the community mean they are overlooked when names are sought for speakers or prizes and the relatively few women are disproportionally asked to sit on committees and participate in other non-research activities at the detriment of their research time. And of course this can be compounded by the broken career patterns and other challenges that stem from child-rearing and family responsibilities.

The under-representation of women at the highest levels of maths has been recognised by learned societies, universities and government resulting in many initiatives, such as the Athena Swan Charter in the UK. Thankfully we now rarely hear of experiences of direct sexism from female mathematicians. Instead for many women, particularly young female mathematicians, there remains an uneasiness about being one of a very few women in a maths department. They have spoken of being "the odd one out" or feeling like "the other" in their work places. More role models – high profile women in the mathematical community – are needed both for younger and established female mathematicians.

So we are thrilled that the brilliant mathematician Maryam Mirzakhani has been awarded the Fields Medal at the ICM 2014 in Seoul, Korea – the first women to be so recognised. Mirzakhani's work is on investigating mathematical surfaces and the geometric structures they can have. She has been recognised for her "rare combination of superb technical ability, bold ambition, far-reaching vision, and deep curiosity" which led to "striking and highly original contributions to geometry and dynamical systems".

Now that the IMU has finally recognised a woman for her mathematical achievements with a Fields medal, we hope that more female mathematicians will be nominated and recognised in the future. It seems fitting that Mirzakhani received her prize from a female head of state, President Park Geun-hye of South Korea, in a ceremony led by Ingrid Debauchies, the president of the International Mathematics Union, and mc-ed by Seon-Hee Lin, a professor from Seoul National University. These four women made up a third of the guests on the stage at the opening ceremony – we hope the recognition of women in mathematics stays at at least this level in the future.

The prizewinners and guests on stage at the opening ceremony of ICM 2014

You can read more about Maryam Mirzakhani and the other Fields medallists, and about the ICM 2014 on Plus.