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What is the value of pi ? Indeed, how can we define pi or e? One way is by defining numbers like pi and e as the limits of monotonic sequences. But what sequences? The programme begins by pi. Pi is clearly the area of a disc with unit radius, but there's a problem; how can the area be found without using It? The solution is to approximate the area by the areas of sequences of interior polygons, as Archimedes did. This is demonstrated by a computer animation using polygons with 6, 12 and 24 sides and so on. Having established an approximate value for pi, the programme goes on to discuss the Monotone Convergence Theorem to Justify this process, and then demonstrates the theorem to get an approximation to pi using exterior polygons. Next the Theorem is proved using a computer animation, and is then applied to a sequence (...) which converges to the number e. Equally, the series (...) is also convergent, which gives away to define e for all real x; this is established by using computer animations firstly to plot e for Integer and rational values, for which the series is proved true, and then to establish that the definition is a reasonable one for irrational values of x, too. This programme should be of interest to both students and teachers of pure mathematics at advanced or undergraduate level.
Metadata describing this Open University video programme
Module code and title: M203, Introduction to pure mathematics
Item code: M203; 02A; 1995
First transmission date: 26-04-1987
Published: 1995
Rights Statement:
Restrictions on use:
Duration: 00:24:40
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Producer: Pip Surgey
Contributors: Phil Rippon; Robin Wilson
Publisher: BBC Open University
Keyword(s): Area; Circle; e (The number).; Monotone Convergence Theorem; Polygon; Least Upper Bound; Exponential Function
Subject terms: Logarithmic functions; Pi
Master spool number: DOU8015
Production number: FOUM476H
Videofinder number: 2279
Available to public: no