Plus Blog

February 5, 2013

How would you go about adding up all the integers from 1 to 100? Tap them into a calculator? Write a little computer code? Or look up the general formula for summing integers?

Legend has it that the task of summing those numbers was given to the young Carl Friedrich Gauss by his teacher at primary school, as a punishment for misbehaving. Gauss didn't have a calculator or computer, no one did at that time, but he came up with the correct answer within seconds. Here's how he did it.

Notice that you can sum the numbers in pairs, starting at either end. First you add 1 and 100 to get 101. Next it's 2 and 99, giving 101 again. The same for 3 and 98. Continuing like this, the last pair you get is 50 and 51 and they give 101 again. Altogether there are 50 pairs all adding to 101, so the answer is 50 x 101 = 5050. Easy — if you're Gauss.

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January 31, 2013
Usain Bolt

Usain Bolt celebrates his victory over 100m and new world record at the Beijing Olympics. Image: Jmex60.

Sometimes you just can't argue with the evidence. If a large sample of very ill people got better after dancing naked at full moon, then surely the dance works. Less contentiously, if the country's best-performing schools produce worse results over time, then surely something is wrong with the education system.

But hang on a second. Before you jump to conclusions, you need to rule out a statistical phenomenon called regression to the mean. The idea is that if you choose a set of measurements because they are quite extreme, and then do the same measurements a little while later, the result is likely to be less extreme.

Think of Usain Bolt. If you measured his performance over 100m the day after he ran a world record, the time you'd get would probably not be another world record, but something slower. This is because his world record performance on the previous day was not entirely down to his physical ability but also to all sorts of other factors - his mood, the condition of the track, the passion of the crowd - which to all intents and purposes are random. On his next run some or all of these factors are probably absent, so his performance will be closer to his personal average, or mean.

Similarly, if you select a group of very ill people to test a drug (or dance) on, then on your next measurement they are likely to feel better simply because their ill-being has regressed to the mean (never mind the placebo effect). You can't automatically assume it was because of the drug. And if you select a group of schools because of their outstanding performance, you're likely to see worse results next time around. You can't necessarily blame the government.

Regression to the mean was first noticed by a cousin of Charles Darwin, Sir Francis Galton, in the 19th century. You can read about him and his discovery in the maths magazine The Commutator. And for an example of how measurements of school performance can potentially give misleading results, read this article by Daniel Read of Warwick Business School.

January 16, 2013

Stephen Hawking was once told by an editor that every equation in a book would halve the sales. Curiously, the opposite seems to happen when it comes to research papers. Include a bit of maths in the abstract (a kind of summary) and people rate your paper higher — even if the maths makes no sense at all. At least this is what a study published in the Journal Judgment and decision making seems to suggests.

Mathematical formulae written on paper

Maths: incomprehensible but impressive?

Kimmo Eriksson, the author of the study, took two abstracts from papers published in respected research journals. One paper was in evolutionary anthropology and the other in sociology. He gave these two abstracts to 200 people, all experienced in reading research papers and all with a postgraduate degree, and asked them to rate the quality of the research described in the abstracts. What the 200 participants didn't know is that Eriksson had randomly added a bit of maths to one of the two abstracts they were looking at. It came in the shape of the following sentence, taken from a third and unrelated paper:

A mathematical model $(T_{PP}=T_0-fT_0d_ f^2−fT_ Pd_ f)$ is developed to describe sequential effects.

That sentence made absolutely no sense in either context.

People with degrees in maths, science and technology weren't fooled by the fake maths, but those with degrees in other areas, such as the humanities, social sciences and education, were: they rated the abstract with the tacked-on sentence higher. "The experimental results suggest a bias for nonsense maths in judgements of quality of research," says Eriksson in his paper.

The effect is probably down to a basic feature of human nature: we tend to be in awe of things we feel we can't understand. Maths, with its reassuring ring of objectivity and definiteness, can boost the credibility of research results. This can be perfectly legitimate: maths is a useful tool in many areas outside of hard science. But Eriksson, who moved from pure maths to interdisciplinary work in social science and cultural studies, isn't entirely happy with the way it is being used in these fields. "In areas like sociology or evolutionary anthropology I found mathematics often to be used in ways that from my viewpoint were illegitimate, such as to make a point that would better be made with only simple logic, or to uncritically take properties of a mathematical model to be properties of the real world, or to include mathematics to make a paper look more impressive," he says in his paper. "If mathematics is held in awe in an unhealthy way, its use is not subjected to sufficient levels of critical thinking."

You can read Eriksson's paper here. There is also an interesting article on this and other bogus maths effect in this article in the Wall Street Journal.

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December 24, 2012

Without doubt the biggest event in physics and maths this year was the discovery of the Higgs boson. Relive the excitement and understand what it's all about with these Plus articles.

Merry Christmas!

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The Higgs boson: A massive discovery — If it looks like the Higgs... and it smells like the Higgs... have we finally found it? Most physicists agree it's safe to say we've finally observed the elusive Higgs boson. And perhaps that is not all....


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Particle hunting at the LHC — Our favourite particle physicist, Ben Allanach, explains exactly what they are looking for at the LHC. Welcome to the world of quantum jelly....


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Secret symmetry and the Higgs boson: Part I and Part II — The notorious Higgs boson, also termed the god particle, is said to have given other particles their mass. But how did it do that? In this two-part article we explore the so-called Higgs mechanism, starting with the humble bar magnet and ending with a dramatic transformation of the early Universe.


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Countdown to the Higgs? — What does all this talk about sigma levels mean? It turns out that finding the Higgs is not so much a matter of catching the beast itself, but keeping a careful count of the evidence it leaves behind.


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Hooray for Higgs! — The LHC gave particle physicists an early Christmas present last year – the first glimpses of the Higgs boson.



December 23, 2012
Timothy Lanzone

Timothy Lanzone on the set.

Travelling Salesman is an unusual movie: despite almost every character being a mathematician there's not a mad person in sight. Moreover, the plot centres on one of the greatest unsolved problems in mathematics, does P = NP? Last month we were lucky enough to host the UK premiere of this movie, here at the Centre for Mathematical Sciences , the home of Plus. We spoke to Jonathan Oppenheim from University College London about the maths behind the movie and to the film's writer and director, Timothy Lanzone about creating drama from mathematics.

You can listen to these interviews in our podcast and read more about the P versus NP problem in this article.


December 22, 2012

Today would have been the 125th birthday of the legendary Indian mathematician Srinivasa Ramanujan. This self-taught genius formed a remarkable working relationship with the mathematician G.H. Hardy which served as inspiration for the 2008 play A disappearing number by Complicite. Read our article on the play and some of the maths behind it, our interview with an actor/mathematician involved in the play, and an article featuring one of Ramanujan's contribution to number theory.


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