Available online
Description
This programme shows how matrix manipulations can help solve an apparently unrelated problem on rabbit breeding.
This programme shows how matrix manipulations can help solve an apparently unrelated problem on rabbit breeding.
| Module code and title: | M101, Mathematics: a foundation course |
|---|---|
| Item code: | M101; 20; 1985 |
| First transmission date: | 21-07-1985 |
| Published: | 1985 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:24:40 |
| + Show more... | |
| Producer: | Andrew Barker |
| Contributors: | Mike Crampin; Phil Rippon |
| Publisher: | BBC Open University |
| Keyword(s): | Eiganlines; Eiganvalues; Fibonacci series; Matrix; Networks; Powers; Rabbits |
| Footage description: | The programme shows how matrix algebra can be used to solve a famous mathematical problem first posed by the 13th century Italian mathematician, Fibonacci. The problem concerns pairs of rabbits which breed and produce new pairs. How many pairs of rabbits will there be in any subsequent month ? The presenter begins by carrying out a simulation of the problem. This leads to a network representation of the process. An animation of the network carries on the simulation and develops the situation for a general month. A formula is produced which gives the number of rabbit pairs in any month in terms of the numbers in previous months. This leads to the series of numbers known as the Fibonacci series 1,1,2,3,5,8,13 The presenter then sets up a matrix representation of the network and an animation develops a matrix representation for the nth month. However, this involves calculating the nth power of the matrix. The presenter finds the eigenvalues and eigenlines of the matrix and then develops the QDQ-1 representation. Using this representation he is able to calculate the nth power and so obtain an explicit formula for the number of pairs of rabbits in the nth month. From the formula it is observed that the ratio of successive terms approaches 1.618.... The presenter then investigates the geometry of the matrix transformation involved and finally uses the geometry to determine that no matter what starting value had been used the ratio of successive terms would always approach 1.618.... |
| Master spool number: | HOU5061 |
| Production number: | FOUM222X |
| Videofinder number: | 2479 |
| Available to public: | no |
