Fibonacci numbers

I researched a scientist or rather a mathematician that made contributions to his discipline such that they have affected a majority of the people that have lived on this earth since his time. His name is Leonardo Pisano. It is hypothesized that Leonardo was born in the town of Pisa which is in modern day Italy circa 1170. Leonardo moved at a young age with his father to a town in northern Algeria. Leonardo’s father held a diplomatic post where his job was to represent the merchants of the republic.

At a young age Leonardo worked with numbers learning the in and outs of accounting and balancing books. In Algeria and other countries that he visited with his father he learned different numbering systems and how they had advantages to the one that he grew up with. In Algeria from the Arabs he learned the base 10 system and was responsible to spreading this system across Europe which in turn was spread across the world and is now the most widely used number system (Connor 1998).

Most people today know Leonardo by his nickname Fibonacci. By the turn of the century Fibonacci had returned to Italy and began to write texts. He wrote on number theory, geometry, algebra, and documented problems and proofs. Fibonacci lived before the printing press had been invented and all copies of his books had to be had written copies from his own hand written copies. Today we still have four of his books; Liber abaci (1202), Practica geometriae (1220), Flos (1225), and Liber quadratorum.

According to an article by Keith Devlin, Executive Director of the Center for the Study of Language and Information at Stanford University, Fibonacci’s first book Liber abaci is “the book that gave numbers to the western world”. Fibonacci was born in the Roman Empire and therefore was taught in his youth the Roman numeral system which is very limiting when one wants to calculate complex equations. As mentioned earlier Fibonacci traveled extensively in northern Africa with his father where he learned the base ten system from the Arabic people who in turn learned it from the people of India who developed it sometime in the first millennium.

In his book Liber abaci or “The Book of calculation” he documented the system in detail that he learned from the Arab traders including its efficiency in performing arithmetic (Delvin 2002). This system included ten symbols including a decimal point which could express partial numbers or fractions (Delvin 2002). The book begins; “These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which in Arabic is called zephirum, any number can be written, as will be demonstrated” (Pisano 1202).

Being the son of a merchant, the book is geared to help trading merchants learn the new system, how to apply it to there trades, and calculate exchange rates (Delvin 2002). It also discusses the rabbit problem for which Fibonacci is most well known. The rabbit problem introduces a sequence in which the next number in the sequence is the sum of the previous two numbers (1, 1, 2, 3, 5, 8, 13, 21). This sequence coincidentally appears randomly in nature in shells, bee hives, and even in electrical resistance which had not even been discovered yet (Knott 2005).

Other books had been written about the Arab-Hindu numbering system but they had more of a mathematic approach and so were mostly only read by serious mathematicians. Since Liber abaci was written to help merchants better their math skills it was much more widely duplicated and introduced the numbering system to common people (Delvin 2002). Fibonacci’s other books were more math oriented and not as popular as Liber abaci.

Practica geometriae dealt primarily with algebra and trigonometry based on the math theory of Euclid. Liber quadratorum (“The book of Squares”) is a book on advance algebra and number theory. One thing that I learned from my research is discussed in this book; all perfect squares can be expressed as a sum of odd integers (f(22)=1+3, f(32)=1+3+5, f(42)=1+3+5+7)(Connor 1998). This to me was amazing and gave a simple look into the brilliant understanding Fibonacci had of the Arab-Hindu number system.

Fibonacci was well respected by his peers and scholar members of the court of the Holy Roman Emperor Frederick II suggested that he meet Fibonacci when his court met in Pisa in 1225. Another of Fibonacci’s books Flos is also largely devoted to algebra and contains answers to a series of problems posed to him in a contest organized for emperor Frederick II. Including an accurate approximation of 10x + 2×2 + x3 = 20 (Connor 1998). I played with this equation for quite awhile with my college math skills and I didn’t come close to an answer.

Unlike me, Fibonacci who learned a majority of his math while studying and learning during his travels with merchants was able to solve this equation with a correct answer out to the 9th decimal. Fibonacci was truly a genius mathematician for his time and I am sure even in our time. Fibonacci was even given an annual salary from the Roman Empire for his contributions to the empire (Connor 1998). I myself am amazed at his understanding of numbers and who knows how many practical applications have been or will be discovered as a result of Fibonacci’s mathematical discoveries.


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