The Fibonacci Sequence
Back to the Middle Ages and the Renaissance
Who was Fibonacci?
Leonardo Fibonacci was born in Pisa, Italy
Pisa was an important commercial town in its day and had links with many Mediterranean ports. Leonardo's father (Guglielmo Bonaccio) was a kind of customs officer in the North African town of Bugia now called Bougie where wax candles were exported to France.
So Leonardo grew up with a North African education under the Moors and later traveled extensively around the Mediterranean coast. He would have met with many merchants and learned of their systems of doing arithmetic. He soon realized the many advantages of the "Hindu-Arabic" system over all the others. He, in his book "Liber abbaci", was one of the first people to introduce the Hindu-Arabic number system into Europe - the positional system we use today - based on ten digits with its decimal point and a symbol for zero.
Fibonacci Sequence
In Fibonacci's book he introduces a problem for his readers to use to practice their arithmetic:
A pair of rabbits are put in a field and, if rabbits take a month to become mature and then produce a new pair every month after that, how many pairs will there be in twelve months time?
To calculate this number, Fibonacci decided to call Fn the number of pairs at the beginning of the nth month. Then F1 =1 and F2 = 2, since at the beginning of the first month there is just the original pair, but at the beginning of the second month the first pair has produced a second pair.
He then noticed that at the beginning of the nth month the pairs can be divided into two groups: a number Fn-1of "old" ones, who were already there after n-1 months; and a number of new ones, who have just been born. Since a new pair becomes fertile after one month and produces its first descendants after one more month, the number of new pairs is equal to the total number of pairs two months earlier, which is Fn-2. As a result,
By using this formula and the initial values F1 =1 and F2 = 2, it is possible to show the number of pairs after one year (233 pairs). This series of numbers Fn is called the Fibonacci sequence. By general agreement, the initial values are normally taken to be 1 and 1 instead of 1 and 2 (so that the following terms in the sequence are shifted):
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, and so on.
