Tuesday, March 18, 2014

The First 12 Numbers in the Fibonacci Sequence

Fibonacci was an Italian mathematician, considered by some as "the most talented western mathematician of the Middle Ages". Leonardo Pisano Bigollo was his name and was also known as Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci and Leonardo Fibonacci. He was best known in Europe for spreading the use of Hindu-Arabic numerical system when his book Liber Abaci (Book of Calculation) was published. Fibonacci sequence was used as an example in this book by introducing it as an exercise involving a population of rabbits in 1202. This sequence of numbers was named after him but he did not discovered it, rather it was already appeared in the natural world that dates back to over two millenia and was first use by Indian mathematicians.

His sample problem:

"A certain man put a pair of rabbits in a place surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair from which the second month on becomes productive?" (Liber abbaci, chapter 12, p. 283-4)
Solution:

At month zero, there is one pair of rabbits. At the beginning of the first month (January 1st), there is one pair of rabbits that has mated, but not yet given birth. At the beginning of the second month, the original pair gives birth, giving rise to another pair of rabbits...and so on. After one full year, there are 377 pairs of rabbits.
The Fibonacci numbers was formed from a recurrent sequence. Beginning with 1, each term of the Fibonacci sequence is the sum of the two previous numbers.

   0+1=1     1+1=2     1+2=3     2+3=5     3+5=8    5+8=13

Fibonacci began the sequence not with 0, 1, 1, 2, as modern mathematicians do but with 1,1, 2, etc.

Hence, the first 12 numbers in the Fibonacci sequence are:

   1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144

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