Available online
Description
This programme explores how mathematicians have calculated the value of Pi down the centuries.
This programme explores how mathematicians have calculated the value of Pi down the centuries.
| Module code and title: | M203, Introduction to pure mathematics |
|---|---|
| Item code: | M203; 03B; 1987 |
| First transmission date: | 30-08-1987 |
| Published: | 1987 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:23:46 |
| Note: | Programme re-versioned in 1995 as FOUM494B |
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| Producer: | David Saunders |
| Contributors: | David Brannan; Phil Rippon; Robin Wilson |
| Publisher: | BBC Open University |
| Keyword(s): | Gregory's series; Pi; Power series; Taylor series |
| Subject terms: | Calculus |
| Footage description: | What's the value of pi? In the Bible it was taken as 3, and many other approximations have been used as well as the famous 3 1/7. Some of these approximations have been found using geometrical constructions, but since the development of the calculus new techniques have evolved which use power series expansions. This programme, which is intended to be viewed in conjunction with Unit 3 of M203 Analysis Block B, gives an historical perspective to the quest for pi and then concentrates on Taylor series expansions involving the inverse tangent functions. The original series (Gregory's series) proves hard to obtain directly, although by means of a clever trick the calculations can be simplified. However, the standard rules which Justify these manipulations (the radius of convergence theorem and the differentiation and integration rules) do not apply in precisely the right way. Nevertheless, the procedure can be justified and the series converges. The trouble is, it converges ridiculously slowly, as a computer demonstration shows. This difficulty was not lost on the early mathematicians who were attempting to calculate pi, and eventually they came up with other formulae which involved the continuation rule for the inverse tangent function as well as power series. Another computer demonstration compares two of these formulae, showing that they give results correct to eight decimal places using only a handful of terms. One such formula, called Machin's formula, has been the basis for computer calculation to an accuracy of 100,000 decimal places, although more recent and even more accurate calculations have used rather different techniques. |
| Master spool number: | HOU5835 |
| Production number: | FOUM282T |
| Videofinder number: | 437 |
| Available to public: | no |
