The Fibonacci Series and The Golden Ratio - PowerPoint PPT Presentation

1 / 14
About This Presentation
Title:

The Fibonacci Series and The Golden Ratio

Description:

A simple numerical series devised by the Italian mathematician ... Nature - flowers and fruit e.g. pineapples, Nautilus Shell. The Great Pyramid of Cheops ... – PowerPoint PPT presentation

Number of Views:2251
Avg rating:3.0/5.0
Slides: 15
Provided by: Ada105
Category:

less

Transcript and Presenter's Notes

Title: The Fibonacci Series and The Golden Ratio


1
The Fibonacci Series and The Golden Ratio
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,
233, 377, 610, 987,
Adam Rowlands Grey College
2
Main Points
  • Background
  • Brief mathematical properties
  • So whys this interesting?

3
Background
What is the Fibonacci series?
  • A simple numerical series devised by the Italian
    mathematician Leonardo de Pisa towards the end of
    the 12th century
  • It was derived from the rabbit problem and was
    described in his most famous book - Liber Abaci
    (Book of the Abacus)
  • A pair of rabbits produces a pair of baby
    rabbits once each month. Each pair of baby
    rabbits requires one month to grow to be adults
    and subsequently produces one pair of baby
    rabbits each month thereafter. Determine the
    number of pairs of adult rabbits some number of
    months after
  • He discovered that the problem could be modelled
    mathematically using the below additive sequence
  • An2 An An1

4
How was this equation derived?
  • The number of adult rabbits in month n2 is the
    combined number of adults and babies from the
    previous month
  • An2 An1 bn1
  • In a given month (say n1) the number of baby
    rabbits is equal to the number of adult rabbits
    in the previous month
  • bn1 An
  • Hence An2 An An1
  • Numerous sequences can be produced using the
    above equation, but the most famous The
    Fibonacci series, uses the root values 0 and 1
  • 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,
    144, 233, 377, 610, 987,

5
Finding a fibonacci number
  • Trivial method of adding previous terms
  • A more sophisticated method involves the number
    Phi (t)
  • Fn 1/t n (-t) - n
  • Binet formula - Jacques Phillipe
  • There are two problems with this method which
    cannot be overcome
  • Phi is an irrational number
  • If n is very large and Fn is required to be a
    precise integer then Phi must have a sufficient
    number of significant digits
  • Hence Fn trunct n /5 0.5

6
Generalised Fibonacci Numbers
  • The most famous series produced using the same
    equation but different root values is the Lucas
    series
  • 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199,
    322, 521, 843
  • A specific Lucas number can be found using a
    similar equation to the Binet formula
  • The Lucas numbers are related to the Fibonacci
    numbers
  • Ln v5Fn

7
How does this relate to Phi?
  • The fundamental link between the Fibonacci series
    and the number phi
  • t limn?8 Fn / Fn-1

8
Mathematical properties of Phi
  • An irrational number defined as (1v5)/2
  • Phi 1.61803
  • No pleasing symmetry
  • Phi appears throughout nature and art
  • t v(1 t)
  • Thus
  • t v(1 v(1 v(1 v(1 v(1 v(1 ..))

9
Where does Phi occur?
  • The Great Pyramid of Cheops
  • Nature - flowers and fruit e.g. pineapples,
    Nautilus Shell

10
The Great Pyramid of Cheops
11
Examples of the Fibonacci series
  • Pascals triangle
  • Nature
  • Pinecones
  • Sunflowers
  • Petals on flowers

12
How does this relate to teaching?
  • Field work
  • Different way of looking at mathematics
  • Historical viewpoint
  • Mystery surrounding numbers
  • Studying famous mathematicians
  • Project

13
Summary
  • The Fibonacci series and phi are some of the
    oldest concepts in mathematics
  • They have been surrounded by intrigue and mystery
    for centuries e.g. Leonardo Da Vinci
  • They exhibit numerous unexpected relations with
    various aspects of mathematics and the natural
    world
  • Possible applications to teaching

14
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,
233, 377, 610, 987,
The Fibonacci Series and The Golden Ratio
Adam Rowlands Grey College
Write a Comment
User Comments (0)
About PowerShow.com