The Fibonacci SeriesEdit
The Fibonacci series of numbers begins as follows:
- 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987...
Each term in the sequence is the sum of the two before it.
Each term can be labelled, according to its place in the series, as F_{0}, F_{1}, F_{2}, F_{3},...
Thus, F_{0} = 0, F_{1} = 1, F_{2} = 1, F_{3} = 2, F_{4} = 3, F_{5} =5, F_{6} = 8, F_{7} = 13, and so on.
Once F_{0} and F_{1} have been defined respectively as 0 and 1, each subsequent term can be defined by the formula:
- F_{n} = F_{n-2} + F_{n-1}
The Fibonacci series is named for Leonardo Fibonacci, variously known as Leonardo of Pisa, Leonardo Pisano and Leonardo Bonacci, the son of a wealthy Italian merchant who ran a trading post in a port in what is now Algeria. Young Fibonacci did a good deal of study under the leading mathematicians of the time, and produced a book of arithmetic called Liber Abaci. In this book, he introduced to Europe the Hindu-Arabic system of numeral notation, much less cumbersome than the old system of Roman numerals, and the basis of our modern numeral system using the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0. Although this is probably his most significant contribution to European mathematics (and European culture as a whole), he is remembered almost exclusively for the series of numbers that now bears his name.
In the Liber Abaci, Fibonacci poses the following problem: beginning with a single pair of (newborn) rabbits, if every pair produces another pair when they two months old, and then a pair every month after that, how many pairs of rabbits will there be after n months? Taking the birth of the original pair as the first month, the monthly totals are 1, 1, 2, 3, 5, 8, 13, 21, and so on. The sequence was known to Indian mathematicians as early as the 6th century, but it was Fibonacci who introduced it to Europe.
The Fibonacci Series |
Fibonacci and the golden ratio | Fibonacci series and Lucas series |