23 December 2009

The Fabulous Fibonacci Numbers

This post is to inspire teachers and students how beatifull Fibonacci numbers. There aren’t many themes in mathematics that permeate more branches of mathematics than the Fibonacci numbers. They come to us from one of the most important books in Western history. This book, Liber Abaci, written in 1202 by Leonardo of Pisa, more popularly known as Fibonacci (1180–1250), * or son of Bonacci, is the first European publication using the Hindu–Arabic numerals that are the basis for our base 10 number system. This alone would qualify it as a landmark book. However, it also contains a "harmless" problem about the regeneration of rabbits. It is the solution of that problem that produces the Fibonacci numbers.

You might have your students try to set up a chart and solve the problem independently before progressing further. It may be stated as follows:

How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair, which becomes productive from the second month on?

It is from this problem that the famous Fibonacci sequence emerged. If we assume that a pair of baby (B) rabbits matures in one month to become offspring-producing adults (A), then we can set up the following chart :

fibonacci pairs

If we let fn be the nth term of the Fibonacci sequence, then :

fibonacci sequence

That is, each term after the first two terms is the sum of the two preceding terms.

Your students may (rightly) ask at this point, What makes this sequence of numbers so spectacular? For one thing, there is a direct relationship between (believe it or not) it and the Golden Section! Consider successive quotients of the Fibonacci numbers:

fibonacci numbers

Furthermore, you can refer students to notice that successive powers of * present us with the Fibonacci numbers.

fibonacci units

If, by now, the students didn’t see the connection, highlight the coefficients and the constants. This is quite incredible; two completely (seemingly) unrelated things suddenly in close relationship to one another. That’s what makes mathematics so wonderful!

* Fibonacci was not a clergyman, as might be expected of early scientists; rather, he was a merchant who traveled extensively throughout the Islamic world and took advantage of reading all he could of the Arabic mathematical writings. He was the first to introduce the Hindu–Arabic numerals to the Christian world in his Liber abaci (1202 and revised in 1228), which first circulated widely in manuscript form and was first published in 1857 as Scritti di Leonardo Pisano (Rome: B. Buoncompagni). The book is a collection of business mathematics, including linear and quadratic equations, square roots and cube roots, and other new topics, seen from the European viewpoint. He begins the book with the comment: “These are the nine figures of the Indians 9 8 7 6 5 4 3 2 1. With these nine figures, and with the symbol 0, which in Arabic is called zephirum, any number can be written, as will be demonstrated below”. From here on, he introduces the decimal position system for the first time in Europe. (Note: The word “zephirum” evolved from the Arabic word as-sifr, which comes from the Sanskrit word, used in India as early as the fifth century,“sunya,” referring to empty.)

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