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.
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!
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!
"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
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 (see here for some articles we have produced in
this context). 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.