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source: GreshamCollege 2011年8月10日

A series of six Mathematics lectures to illustrate the mathematics underpinning our everyday lives. Professor John D. Barrow gave these lectures during 2010-11 as the Gresham Professor of Geometry. All information about these lectures and all future ones can be found on the Gresham College website: http://www.gresham.ac.uk

The transcript and downloadable versions of the lecture are available from the Gresham College website: http://www.gresham.ac.uk/lectures-and...

1 1:04:15 Maths with Pictures

How pictures have been used in mathematics. The use of illustrations in ancient mathematics books, the invention of the first graphs and the representation of probabilities, sets and formulae by pictures. We look at the role played by computers in exploring and displaying the behaviour of extremely large and complicated problems. This has changed the culture of applied mathematics and science and influences the way research is done and the forms in which it is presented.

2 1:03:17 Continued Fractions

What are continued fractions? How can they tell us what is the most irrational number? What are they good for and what unexpected properties do they possess? How did Ramanujan make good use of their odd features to make striking discoveries? We will look at how they have played a role in the study of numbers, chaos, gears and astronomical motions.

3 1:01:22 Mathematics and the Bounce of the Superball

The commercially available 'Superball' of hard rough rubber displays many counterintuitive properties which seem to violate Newton's laws of motion. We will see that the Superball can be understood but its behaviour is completely different to a billiard ball when it undergoes collisions with a wall. We will look also at some other unusual motions of swerving and spinning balls in sports.

4 56:15 The Uses of Irrationality: Paper Sizes and the Golden Ratio

Is there anything mathematically interesting about the paper sizes we use? We will see that their range of sizes has special features that facilitates their use in Xerox machines. The standard US Letter system of sizes is different and creates problems when you want to reduce copies in size. These examples will lead us towards the special properties of certain mathematical ratios in maths, science and art.

5 1:01:22 Benford's Very Strange Law

The first digits of randomly chosen numbers arising naturally or in human affairs display surprising statistical regularities. We will see why this distribution of digits, first found by Simon Newcomb and Frank Benford, is so ubiquitous and how it has been used to check for fraudulent accountancy and for suspicious vote counts in some national elections.

6 59:20 Doing Business in Interstellar Space

Imagine that interstellar trade is possible at speeds close to the speed of light. It must incorporate the insights of Einstein's special theory of relativity, which teaches us that clocks on board a spaceship moving at high velocity will ensure time at different rates relative to clocks at the point of departure. This means that time travel into the future is possible. Which time-keeping should we use? What would happen to economics if time-travel to the past was also possible?

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