Suppose you have people in a room and each person shakes hands with each other person once. How many handshakes do you get in total? The first person shakes hands with other people, the second shakes hands with the remaining people, the second shakes hands with remaining people, etc, giving a total of
But we can also look at this in another way: each person shakes hands with others and there are people, giving handshakes. But this counts every handshake twice, so we need to divide by 2, giving a total of
Putting these two arguments together, we have just come up with the formula for summing the first integers and we’ve proved that it is correct:
At 9:59 pm (UK time) on Friday, May 31, 2013, asteroid 1998 QE2 will sail serenely past Earth, getting no closer than about 3.6 million miles (5.8 million kilometers), or about 15 times the distance between Earth and the Moon, its closest approach for at least the next two centuries. And while QE2 is not of much interest to those astronomers and scientists on the lookout for hazardous asteroids, it is of interest to those who dabble in radar astronomy and have a 230-foot (70-meter) – or larger – radar telescope at their disposal. But you can also catch a glimpse of the asteroid by the power of the internet.
NASA is showing live telescope images of the asteroid and hosting a discussion with experts at 6.30 pm (UK time) today, Thursday 30 May, on NASA TV. You can even submit questions in advance to @AsteroidWatch on Twitter with the hashtag #asteroidQE2.
Is there a perfect voting system? In the 1950s the economist Kenneth
Arrow asked himself this question and found that the answer is no, at
least in the setting he imagined.
Kenneth defined a voting system as follows. There is a population of
voters each of whom comes up with a preference ranking of the candidates.
A voting system takes these millions of preference rankings as input and
by some method returns a single ranking of candidates as output. The
government can then be formed on the basis of this single ranking.
For a voting system to make any democratic sense, Kenneth required it to
satisfy each of the following, fairly basic constraints:
The system should reflect the wishes of more than just one individual
(so there's no dictator).
If all voters prefer candidate x to candidate y, then x should come
above y in the final result (this condition is sometimes called unanimity).
The voting system should always return exactly one clear final
ranking (this condition is known as universality).
He also added a fourth, slightly more subtle condition:
In the final result, whether one candidate is ranked above another,
say x above y, should only depend on how individual voters ranked x
compared to y. It shouldn't depend on how they ranked either of the two
compared to a third candidate, z. Arrow called this condition independence
of irrelevant alternatives.
Arrow proved mathematically that if there are three or more candidates
and two or more voters, no voting system that works by taking voters'
preference rankings as input and returns a single ranking as output can
satisfy all the four conditions. His theorem, called Arrow's
Impossibility Theorem helped to earn him the 1972 Nobel Prize in
From the makers of Dimensions comes a great free online movie exploring dynamical systems, the butterfly effect and chaos theory by means of stunning visuals accompanied by a beautiful musical score. The dynamics of the weather, the three-body problem, Smale's horseshoe — it's all there, including cute lego athletes whizzing around a Lorenz attractor.
"The film is for everybody," says the mathematician Étienne Ghys, one of its creators. "It is split into nine chapters. The level [of the mathematics] continuously increases from chapter one to chapter nine and, in principle, one should not be frustrated if one does not watch the film until the very end." So whatever your level of maths there is something in it for you.
The field of dynamical systems was created by Henri Poincaré some 125 years ago. "Its purpose is to understand the motion of mechanical systems, like celestial bodies for instance, without solving any equations," says Ghys. "The trick is to use pictures to prove theorems: this is just what is needed for a movie."
Poincaré was indeed something of a pioneer in the use of pictures. "During the nineteenth century, especially in France, there was a period in maths during which pictures were considered a taboo," says Ghys. "They were supposed to be evil: the origin of mistakes. Under the influence of mathematicians like Poincaré mathematics, or at least some parts of maths, has become more visual and movies are indeed a wonderful tool to explain things. One picture is worth a thousand words!"
Every year the London Mathematical Society (which despite its name represents academic mathematicians throughout the UK) puts on a pair of popular lectures that are well worth seeing. This year's speakers will be Ray Hill, showing how the
misuse of mathematics can lead to
miscarriages of justice and how to prevent them, and a good friend of Plus, Vicky Neale, talking about intriguing problems to do with adding numbers and their elegant solutions (if they exist!).
The lectures will take place in London on 25th June 2013 at 7pm and in Birmingham on 20th September 2013 at 6:30pm. Admission is free but you need to register for a ticket, either online or by emailing firstname.lastname@example.org.
John D. Barrow FRS has been a Professor of Mathematical Sciences at the University of Cambridge since 1999, carrying out research in mathematical physics, with special interest in cosmology, gravitation, particle physics and associated applied mathematics. He is the author of over 420 articles and 19 books, translated in 28 languages, exploring the wider historical, philosophical and cultural ramifications of developments in mathematics, physics and astronomy. Most importantly (to us anyway), he is the director of the Millennium Mathematics Project of which Plus is a part.
The lecture starts at 7:30pm and you can book tickets here (£8/£6 concession).