## Key Concepts: Mathematics - Fibonacci Numbers

What is it?

The Fibonacci sequence of whole numbers is:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584,...

Have you figured out the pattern? What is the next number in the sequence? Answer below

The sequence is widely known for its many intriguing properties. Here are but a few.
• Every term is the sum of the previous two
• If you add the terms of the sequence then the result of each of these sums will form a sequence as well
1+1=2
1+1+2=4
1+1+2+3=8
1+1+2+3+5=12
1+1+2+3+5+8=20
....
• The addition of n terms of the Fibonacci sequence turns out to be 1 less than the next but one Fibonacci number.
• The Golden Ratio
• If a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates φ (the Golden Ratio)
• For example 987/610 ≈ 1.6180327868852
• These approximations are alternately lower and higher than φ, and converge on φ as the Fibonacci numbers increase
• It is found in nature
• The number of sprials formed from the number of seeds in the spirals of sunflowers
• For example: 34 in one direction, 55 in the other
•  It is found in manmade creations
• The room proportions and building proportions designed by architects
• In music by Classical composers
• For example: Bartok's Dance Suite
Origins

Leonardo of Pisa (also known as Fibonacci) introduced the sequence to Western European mathematics in his 1202 book Liber Abaci. The numbers had been previously described in Indian mathematics. In the book he considers the growth of an idealised (biologically unrealistic) rabbit population, assuming that: a newly-born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?

Conclusion: At the end of the nth month, the number of new pairs of rabbits is equal to the number of pairs in month n-2 plus the number of pairs alive last month. This is the nth Fibonacci number.

Books

Resource: Henderson, M., Baker, J., & Crilly, T. (2009). 100 most important science ideas : key concepts in genetics, physics and mathematics. Buffalo, N.Y. : Firefly Books.