MATHEMAGICA
 
Interesting mathematical snippets from the history and practice of mathematics
 
July/August instalment
Fibonacci and his numbers
Fibonacci, or Leonardo of Pisa, lived around 1175 to 1250
 
 
 
  He is widely known for the sequence of numbers   
      1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...... 
 (He contributed to mathematics in a number of   
 other ways ..) The nth term in the sequence is the sum of the previous two terms 
               i.e.  Tn  =  Tn-1  +  Tn-2 
   This is often presented as a story about the   
breeding of pairs of rabbits, which are presumed to become foecund in their third year, to breed  
once per year thereafter yielding another pair of   rabbits each time.  
 
 The rabbits live for as many years as the required  
 number of terms in the sequence 
 For example after 4 years there is the original pair  
 plus their offspring from year 3 and year 4, i.e  
 three pairs. After 5 years  two breeding pairs of  
 rabbits exist, giving two pairs of offspring for this  
 year, plus a juvenile pair from the previous year. A total of  five pairs, 5 being the fifth Fibonacci  
 number.
 
<-------------------------------mature----------------------------------->   <------Juvenile------> 
Year 8 Year 7 Year 6 Year 5 Year 4 Year 3 Year 2 Year 1 Total No. prs
0 0 0 0 0 0 0 1 1
0 0 0 0 0 0 1 0 1
0 0 0 0 0 1 0 1 2
0 0 0 0 1 0 1 1 3
0 0 0 1 0 1 1 2 5
0 0 1 0 1 1 2 3 8
0 1 0 1 1 2 3 5 13
1 0 1 1 2 3 5 8 21
 
Fibonacci  numbers represent approximations to an exponential growth curve. For high n  the approximations Tn are closer to a smooth curve. The ratio of successive terms  Tn+1/Tn approaches ~1.618 (the golden ratio).
(Try it for yourself to see that the ratios rapidly approach that value).
An approximation to the curve is y = 0.724 e0.4811(n-1) , n = 1, 2, 3 ....
(e0.4811 = 1.618)
(Find a better approximation )
 
 
 
For more information on Fibonacci and his numbers return to this page in September
 
 
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