Available online
Description
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 consider...ing 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.
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 consider...ing 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.
| 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 |
| + Show more... | |
| 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 |
