The Fabulous Fibonacci Numbers

By AlFred S. Posamentier, Ph.D.

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.

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 sentence: “these are the nine figures of the Indians 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 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” evolves 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.)*

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 is from this problem that the famous *Fibonacci sequence* emerged. The problem 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?*

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: **(See Chart I Below)**

The number of pairs of mature rabbits (Adults) living each month determines the Fibonacci sequence (column of A):

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, …

If we let *f*n be the nth term of the Fibonacci sequence, then

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 number so spectacular. You can fortify yourself by preparing some of the many applications of the Fibonacci numbers in nature, architecture, economics, art, and, of course, other branches of mathematics! It would be artificially modest of me not to suggest my book to you as a rather comprehensive source to find out more about this magnificent ubiquitous numbers. This book is readily available in book shops or on-line: *The Fabulous Fibonacci Numbers*, by Alfred S. Posamentier and Ingmar Lehmann (Prometheus Books, 2007).

For one, there is a direct relationship (believe it or not) between the Fibonacci numbers and the Golden Section! Consider successive quotients of the Fibonacci numbers:

N.B. the bar above some of the numbers above means that these repeat indefinitely.

The golden section has an approximate value of 1.6180339887498948482045868343656…, which is gradually being approached by successive quotients of successive Fibonacci Numbers. As you read more about these endlessly charming numbers, you will get a better feeling for the beauty of mathematics!

*Dr. Alfred S. Posamentier is Dean of the School of Education at City College of NY, author of over 40 Mathematics books including: “*Math Wonders to Inspire Teachers and Students” *(ASCD, 2003) and “*The Fabulous Fibonacci Numbers*” (Prometheus, 2007), and member of the NYS Mathematics Standards Committee.*