Plus Blog

February 17, 2010
Wednesday, February 17, 2010

Maths in a minute — Achilles and the tortoise

Achilles and a tortoise are competing in a 100m sprint. Confident in his victory, Archilles lets the tortoise start 10m ahead of him. The race starts, Achilles zooms off and the tortoise starts bumbling along. When Achilles has reached the point A from where the tortoise started, it has crawled along by a small distance to point B. In a flash Achilles reaches B, but the tortoise is already at point C. When he reaches C, the tortoise is at D. When he's at D, the tortoise is at E. And so on. He's never going to catch up with the tortoise, so he has no chance of winning the race.

Something's wrong here, but what? Let's assume that Achilles is ten times faster than the tortoise and that both are moving at constant speed. In the times it takes Achilles to travel the first 10m to point A, the tortoise, being ten times slower, has only moved by 1m to point B. By the time Achilles has travelled 1m to point B, the tortoise has crawled along by 0.1m to point C. And so on. After n such steps the tortoise has travelled

1+1/10+1/100+1/1000+ .... +1/10(n-1) metres.

And this is where the flaw of the argument lies. The tortoise will never cover the 90m it has to run using steps like these, no matter how many of them it takes. In fact, the distance covered in this way will never exceed 10/9=1.111... metres. This is because the geometric progression

1+1/10+1/100+1/1000+...

converges to 10/9. Since the tortoise is travelling at constant speed, it covers this distance in a finite time, and it's precisely when it's done that that Achilles overtakes it.

This problem is known as one of Zeno's paradoxes, after the ancient Greek philosopher Zeno, who used paradoxes like this one to argue that motion is just an illusion.

and about convergent series in the Plus articles
An infinite series of surprises,
Outer space: Series, and
In perfect harmony.

posted by Plus @ 12:10 PM

February 17, 2010
Wednesday, February 17, 2010

A central prediction of Albert Einstein's general theory of relativity is that gravity makes clocks tick more slowly — time passes slower when you're close to a massive body like the Earth, compared to when you're further away from it where its gravitational pull is weaker. This prediction has already been confirmed in experiments using airplanes and rockets, but a new experiment in an atom interferometer measures the slowdown 10,000 times more accurately than before — and finds it to be exactly what Einstein predicted.

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posted by Plus @ 12:27 PM

At 12:05 PM,  Anonymous said...

So what is the maximum and minimum speed of time in our universe?

At 3:06 PM,  Plus said...

The longest interval of time for some process (eg a heart beat or a human lifetime) is that measured by a clock moving with the observer. It is
called proper time. The length of that interval measured by some other clock in relative motion to that observer is always less than the proper time and as the relative speed approaches that of light, it tends to zero.

The twin paradox (see http://plus.maths.org/issue36/features/aiden) is related. If a twin stays at home and lives for 10 years on his watch while his identical twin goes off on a spacetrip at near light speed, then the travelling twin will return to find that he is
younger than his stay-at-home twin when he is reunited with him on earth.

So the maximal time is given by the proper time measured by a clock moving with you, and the minimum time can be arbitrarily small as the relative speed approaches that of light, or the gravitational field approaches the value needed to
make a black hole.

At 8:31 PM,  mdharley said...

There is no such thing as the General Theory Of Relativity. Einstein was successful in developing the Special Theory Of Relativity which seems to describe much of what we know about our own universe in accord with conditions that he defined in the theory. He was unsuccessful in deriving the General Theory Of Relativity which, if ever derived, would apply under any set Of conditions. The Special Theory Of Relativity is to the Generla Theory Of Relativity as a Square is to a Polygon.
mdharley@hotmail.com

February 17, 2010
Wednesday, February 17, 2010

Browse with Plus: Symmetry, reality's riddle

Marcus du Sautoy is a mathematician and Charles Simonyi Professor for the Public Understanding of Science. In this TED talk he explores how the world turns on symmetry — from the spin of subatomic particles to the dizzying beauty of an arabesque — complete with an introduction to groups.

Marcus du Sautoy has also written several articles for Plus:

posted by Plus @ 12:23 PM

February 12, 2010
Friday, February 12, 2010

Maths at the Cambridge Science Festival

If you're wondering how to feed your maths habit between the 8th and 21st of March, then why not head to Cambridge for the 2010 Cambridge Science Festival? There'll be plenty of free maths events, including:

• IMAGINARY: through the eyes of mathematics — A travelling exhibition of beautiful mathematical images and artwork taken from algebraic geometry and differential geometry in which visitors are able to create their own mathematical art. Age range: 12+.
• Conversations across science and art — A talk and discussion event centred on the relationship between science and art, including the presentation Every picture tells a story by Professor John D Barrow and a talk by Professor Gerry Gilmore exploring the relationship between art and astronomy. Age range: 14+.
• Enigma: codes and codebreaking — The Enigma cipher was one of the most powerful weapons of the Second World War. An apparently unbreakable code. How did a small group of mathematicians crack it? Come and see a demonstration of a genuine Enigma machine, and try your hand at breaking different codes used through 2500 years of history! Age range: 8+.
• Who Wants To Be a Mathionaire? — Explore the maths of probability, chance and uncertainty in this exciting and highly interactive game-show style quiz, using hand-held voting technology to answer against the clock! Age range: 14+.
• The Maths and Physics of Sport — Professor John D Barrow looks at some applications of physics and simple mathematics to a variety of sports, including weightlifting, rowing, throwing, jumping, drag car racing, balance sports, and track athletics, as well as some of the paradoxical systems of judging used in ice skating, and the effects of latitude and air resistance on some performances. Age range: 14+.
• What's the risk of getting out of bed? — We are constantly being exhorted to change our behaviour to reduce the chances that things will turn out badly for us, and government is continually intervening to make our society safer. But are we being too cautious? In this lecture, Professor David Spiegelhalter will look at attempts to measure and communicate the benefits, and possible harms, of risk reduction in a range of areas, from swine flu to climate change, heroin to hang-gliding. Age range: 14+.
• The hands on maths fair — Games and puzzles for all ages from the University's Millennium Mathematics Project. Pit your wits against the SOMA cube, tangrams, Auntie's Tea Cups or giant dominoes, and sharpen your strategic reasoning skills! Age range: 5+.

To find out about all the Cambridge Science Festival events go to the festival website.

posted by Plus @ 1:01 PM

February 10, 2010
Wednesday, February 10, 2010

Unintended consequences of mathematics

Mathematicians (and Plus authors) John Barrow, Colva Roney-Dougal and Marcus du Sautoy will be discussing unintended consequences in mathematics with Melvyn Bragg on his BBC Radio 4 programme In Our Time tomorrow morning at 9am.

Image courtesy NASA

Many of the most exciting developments in science is is when knowledge from one area such as pure mathematics unexpectedly crosses boundaries to provide deeper understanding of a previously unconnected problem in another area. The programme will explore many such unintended consequences, including how the ancient purely geometric study of conic sections turned out to be vital in understanding the orbits of the planets, how Einstein used the theoretical concepts from non-Euclidean geometry for his groundbreaking work on special relativity, and how the number theory provided the security necessary for our digital age.

You can read more from John, Colva and Marcus on Plus, as well as articles on conic sections and planetary orbits, non-euclidean geometry and special relativity, and number theory and chryptography.

posted by Rachel @ 2:09 PM

At 8:02 PM,  David said...

I thought this show was brilliant, and I would like to listen again wish my son who is studying maths A Level. This show really made mathematics come alive.

February 4, 2010
Thursday, February 04, 2010

Putting the magic back into classroom maths

Mathematics and magic may seem a strange combination, but Queen Mary's Matt Parker and Peter McOwan want to show students otherwise. They have produced many of the The Manual of Mathematical Magic, a unique kit of magical miracles, to show that the most powerful magical effects performed today have a mathematical basis.

Freely available to any school in England, the Manual exposes the secrets behind street magic, close-up and stage tricks, revealing the varied and exciting everyday uses for the mathematics powering your magic. It gives young mathematicians the chance to be creative, finding new ways to solve problems and discovering the key to the perfect magic trick. Along the journey they will also uncover the skills of a good mathematician, one with the useful employment skills you get from being good at mathematics.

Both McOwen and Parker regularly visit secondary schools to do Mathematical Magic shows for students. “Our goal is to help more students engage with Mathematics," reveals Parker, who is also involved with the More Maths Grads programme. "Magic tricks get the students excited and then we show them the mathematical principles that make the whole trick hang together. We also reveal how the same Mathematics underpins everything from medical scans to sending text messages.”

As well as The Manual of Mathematical Magic, the kit also contains a pack of cards, notebook and pencil – all of which have hidden Mathematical Magic. Teachers can use the tricks in the book in their lessons and then explain the Mathematics and its applications.

“Maths is magic. But too often school maths is a dull diet which sucks the joy out of what should be a thrilling and beautiful subject," said Paul McGarr, Deputy leader Maths Faculty at Langdon Park School where Parker gave a magical lesson to Year 10 pupils this week. "This new pack, quite literally, helps put the magic back into classroom maths. My pupils really loved it, they were engaged, excited and happy – not bad for last period of a long day! The 'wow' was audible when they saw some of the tricks demonstrated, and you could almost taste their intense curiosity to find out how it was done using maths. I would strongly recommend teachers to get hold of this pack and use it.”