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November 23, 2006
Thursday, November 23, 2006

Mathematical moments - Taking chances with De Moivre

Abraham De Moivre

Born on the 26th of May 1667 in Vitry-le-Francois, France
Died on the 27th of November 1754 in London, England

When De Moivre first came across Newton's famous work the "Principia" he was so struck by its depth and rigour that he immediately bought a copy and cut it into pieces - carrying just a few pages at a time was the only way he could study the work while making his rounds tutoring private students in London.

But it wasn't just dedication that gained him full marks. Since an early age he had been interested in maths, especially in games of chance, and he is today known as a pioneer of probability theory and of analytic geometry. His "Doctrines of chance" presented the broadest and most rigorous treatment of probability of its day, and he is credited with deriving the normal curve and developing the concept of standard deviation. His name is famously attached to a formula that gives geometric meaning to powers of complex numbers by expressing them in terms of trigonometry.

De Moivre's eminence as a mathematician was recognised by many of his most prominent contemporaries, including Newton, who he was friends with, and Leibniz. Interestingly, the Royal Society called upon him to referee Newton and Leibniz's dispute about who had first invented the calculus.

Sadly, though, De Moivre's genius was never rewarded professionally. As a French national who had been expelled from France (after a prison sentence) because of his protestant religion, he remained a foreigner in London. Despite the support of his prominent friends he was never employed by a university. He made a living as a private tutor and died in poverty.

Death played an important role in his mathematics. Together with Halley, who gave his name to the comet, he set about investigating mortality statistics, laying the foundations for actuary theory used by life insurances.

Most curiously, De Moivre is said to have used maths to predict his own death. He had noticed that he was sleeping 15 minutes longer every day. Analysing the arithmetic progression 15, 30, 45, .... , he calculated that on the 27th of November 1754 he would sleep through the full 24 hours. He was right - it was the day he died.

Find out more about De Moivre on the MacTutor history of maths archive, about De Moivre, death and statistics in this past Plus article, and about complex numbers and trigonometry on cut-the-knot.

posted by Plus @ 4:21 PM

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November 21, 2006
Tuesday, November 21, 2006

More or Less back for more

Plus's favourite radio show on all things mathematical is back on the air. More or Less has two series a year on BBC Radio 4, exploring maths in politics, health, avalanches, and much more. In the first shows of this season they have already covered drug testing in sport, the economics of climate change, uncovered the games behind hospital waiting times, and the perverse nature of randomness.

The show is produced in association with the Open University, which provides additional material on their site. You can hear all the past shows online at the More or Less site, and listen live every Monday at 4.30pm on BBC Radio 4.

You can read more about More or Less from presenter Andrew Dilnot in a past article on Plus.

posted by Plus @ 10:41 AM

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November 7, 2006
Tuesday, November 07, 2006

Mathematical moments - The mathematical Bernoullis from Basel

Brothers Jacob I (1654-1705) and Johann I (1667-1748)
Their nephew Nikolaus I (1687-1759)
Johann's sons Nikolaus II (1695-1726), Daniel (1700-1782) and Johann II (1710-1790)
And Johann II's sons Johann III (1744-1807) and Jacob II (1759-1789)

The famous mathematical family, the Bernoullis, produced an astounding eight mathematicians over three generations. The sheer number of them, and the family habit of using the same first names, required the numbering system you see above to keep track of them all. And given the family's mathematical success, you would think that each generation was actively encouraged to study the subject. But instead the mathematical members of the family often had to study mathematics and astronomy against the wishes of their parents. Indeed there were enough quarrels, backstabbing and even untimely deaths among the Bernoullis to script a soap opera.

Jacob (I) Bernoulli was the first member of the family to study maths, and taught his brother Johann (I) who had been forced to study medicine. The brothers worked on similar topics, such as calculus (Jacob was the first mathematician to use the term 'integral') and studying families of curves such as the catenary, the curve of a suspended string. However, in what would prove to be typical behaviour in the family, the brothers soon went from collaborators to rivals, publicly criticising each other's intellect and competing to solve the same mathematical problems.

Jacob and Johann taught their nephew, Nikolaus I, mathematics, and Nikolaus assisted his uncle, Jacob, in publishing his works. Nikolaus is known for posing the probability problem the "St Petersburg paradox", which describes a gambling game that no-one would reasonably play, despite a possibly infinite prize.

Nikolaus's cousin Daniel (son of Johann) provided an explanation of the St Petersburg paradox. Daniel, probably the most famous mathematician of the family, did his most important work on fluid dynamics, and gave the Bernoulli principle. However, continuing the family's bitter history, Daniel had a difficult relationship with his father Johann, who did not want him as a mathematical competitor. Johann tried to stop Daniel from studying mathematics, and even attempted to plagiarise Daniel's greatest work, "Hydrodynamica".

Johann I's other two sons were also mathematicians. His favourite Nikolaus II worked on the problem of trajectories, and the mathematical arguments behind Newton and Leibniz's dispute over who had invented calculus. Johann II worked in mathematical physics.

Johann II also had two mathematical sons. Johann III was a child prodigy, and was just 19 years old when he was appointed to the Berlin Academy. He produced work in astronomy and probability, but his accounts of his travels in Germany had a greater impact historically. Jacob II worked on mathematical physics at the St Petersburg Academy of Sciences, and married Euler's granddaughter. Sadly he drowned in the Neva River when he was only 29 years old.

The Bernoulli family, despite its infighting and bitterness, dominated mathematics in the 17th and 18th centuries. Together with their contemporaries Newton, Leibniz, Euler and Lagrange, they laid many of the foundations of mathematics and physics that we still use today.

"There is no philosophy which is not founded upon knowledge of the phenomena, but to get any profit from this knowledge it is absolutely necessary to be a mathematician." - Daniel Bernoulli

Read more about the Bernoulli clan in the MacTutor History of Mathematics Archive...

And about the Bernoullis, the St Petersburg Paradox and the Bernoulli principle on Plus...
The gentlemen from Basel and the Petersburg Paradox
Daniel Bernoulli and the making of the fluid equation
Understanding turbulence
Testing Bernoulli: a simple experiment

And you can find out more mathematical moments by subscribing to the Plus newsletter!

posted by Plus @ 11:41 AM

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October 2, 2006
Monday, October 02, 2006

Plus new writers award - the entries are in!

Our inaugural writing competition has just closed, and we were very pleased to arrive to overflowing mailboxes — both virtual and real!

Our esteemed panel of judges: best selling authors Marcus du Sautoy and John Barrow, and Economist correspondent and ex-editor of Plus Helen Joyce, will be judging the excellent entries over the next couple of months. The prize pool has swelled over the year and now includes many of the best popular science books, as well as a subscription to nature and an iPod!

So stay tuned for the December issue of Plus when you can find out who won and how they brought mathematics to life!

posted by Plus @ 5:07 PM

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September 25, 2006
Monday, September 25, 2006

Celebrating mathematics at work

The relevance and usefulness of mathematics is most clearly demonstrated in its application to real-world problems, many of which have featured in Plus over the years. To celebrate mathematics at work in the real-world, the International Council for Industrial and Applied Mathematics have announced the winners of the five ICIAM prizes for 2007.

The Pioneer Prize, recognising innovations in applied mathematics, was jointly won by Ingrid Daubechies (Princeton University, USA) and Heinz Engl (Johannes Kepler Universität Linz, Austria). Daubechies' work on wavelets has found widespread use in image processing and frequency analysis. Engl was recognised for his work on inverse problems to the solution of a wide range of industrial problems and for his promotion world-wide of industrial and applied mathematics.

The Collatz Prize is awarded to outstanding applied mathematicians under the age of 42. It was won by Felix Otto (Universität Bonn, Germany) for his fundamental contributions in areas ranging from micromagnetics to mass transportation problems.

Joseph Keller (Stanford University, USA) won the Lagrange Prize for his exceptional contributions to the field. Not only has he influenced the course of modern applied mathematics over the last 50 years, he has also had a great impact in pure mathematics as well. He developed the Geometrical Theory of Diffractions that provided the first systematic description of wave propagation around edges and corners of an obstacle. It has been widely used for radar reflection from targets, elastic wave scattering from defects in solids, acoustic wave propagation in the ocean radar and many other fields. He has also made contributions to quantum mechanics, optics, acoustics, biophysics and biomechanics and transport theory.

Peter Deuflhard (ZIB Berlin, Germany), winner of the Maxwell Prize, is one of the founders of modern scientific computing, and is described as having made a contribution to applied mathematics that is without parallel, including applications in chemical engineering, medicine and biotechnology, microwave technology and nano-optics.

The Su Buchin Prize is more unusual in the field of mathematics, recognising the "application of Mathematics to emerging economies and human development". The winner, Gilbert Strang (MIT, USA), is one of the most recognised mathematicians in the developing world through his promotion of mathematical research and education across the globe. As well as numerous visits himself, he has organised visits by other mathematicians to developing countries, and his educational materials are available on the web, free-of-charge to any user anywhere in the world, through MIT's OpenCourseWare. Alongside his contributions in bringing mathematics to people around the world, his research has made contributions in many areas of pure and applied mathematics.

The prizes will be presented at the International Congress for Industrial and Applied Mathematics in Zürich next year, a major international celebration of mathematics in action. More information on the prizes, and the winners, can be found at the ICIAM website. And you can read more about industrial mathematics on Plus.

posted by Plus @ 5:04 PM

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September 19, 2006
Tuesday, September 19, 2006

Einstein passes with 99.5% result

On September 14th, after 3 years observing a rare double pulsar, an international team led by Prof. Michael Kramer from the Jodrell Bank Observatory announced staggering results confirming Einstein's general theory of relativity to 99.5% accuracy. The observations were made on a pair of pulsars orbiting each other approximately 2000 light years from Earth. Each pulsar weighs a little more than the Sun but has a diameter of around 10km. The extreme mass of the pair gives us a rare opportunity to study the predicted effects of general relativity.

Sometimes when a large star collapses it becomes a very small, very dense object known as a neutron star. A pulsar is a rotating neutron star. Pulsars are extremely useful for astronomical observations as they emit constant, powerful beams of radio waves similar to how a lighthouse emits a beam of light. Each time a beam sweeps past Earth we observe a distinctive radio pulse. By precisely measuring the time between pulses we can make detailed calculations on the effect gravity is having on the radio waves. By current theory, this behaviour should be governed by the laws of general relativity and it turns out that the predictions match with observation to an astonishing level of accuracy.

Another very important result is that the distance between the two pulsars is shrinking by around 7mm per day. This also agrees with Einstein's predictions. According to general relativity, the pulsars should be emitting gravitational waves and although these waves have never been directly detected, their emission should cause the pulsar system to lose energy. The observable effect of this would be the pulsars spiraling towards each other by precisely the amount observed giving compelling evidence for the existence of gravitational waves.

However Dr. Kramer believes that there are still many more exciting results to come: "The double pulsar is really quite an amazing system. It not only tells us a lot about general relativity, but it is a superb probe of the extreme physics of super-dense matter and strong magnetic fields but is also helping us to understand the complex mechanisms that generate the pulsar's radio beacons." He concludes; "We have only just begun to exploit its potential!"

You can read the full story at the Jodrell Bank Observatory website.

posted by Plus @ 2:06 PM

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