The Fibonacci Series and The Golden Ratio

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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
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Main Points

• Background
• Brief mathematical properties
• So whys this interesting?

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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

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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,

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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

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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

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How does this relate to Phi?

• The fundamental link between the Fibonacci series
  and the number phi
• t limn?8 Fn / Fn-1

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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 ..))

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Where does Phi occur?

• The Great Pyramid of Cheops
• Nature - flowers and fruit e.g. pineapples,
  Nautilus Shell

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The Great Pyramid of Cheops
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Examples of the Fibonacci series

• Pascals triangle

• Nature
• Pinecones
• Sunflowers
• Petals on flowers

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How does this relate to teaching?

• Field work
• Different way of looking at mathematics
• Historical viewpoint
• Mystery surrounding numbers
• Studying famous mathematicians

• Project

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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

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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