Saturday, May 31, 2014

Daily 8 and Weekly 4: Fibonacci

Fibonacci (History of Math, compiled using Wikipedia articles and handouts from class.)

Leonardo Pisano Bigollo, more popularly known as Fibonacci, was an Italian mathematician born circa 1170. His most famous work, Liber Abaci (no, not Liberace), was one of the first texts in mathematics to use the Hindu-Arabic number system that we use today. Before this system, Roman numerals were used which, as we experienced in class, was a relatively difficult system to use for computations. So, Fibonacci introduced the new Hindu-Arabic system with his book, and provided explanations of how it is used. One such example was the division of 18456 by 17. This example seemed appropriate, since it allowed Fibonacci to illustrate the way place values were used in the new system, along with the ease of using this system to make relatively complex computations.

In addition to introducing a new number system, Fibonacci included sections covering other topics in his Liber Abaci. After the number system section came a section that included examples of the system as used in commerce, like calculating profit and interest, and converting currency. Next was a section containing mathematical problems. The most famous of these problems, concerning the growth of a population of rabbits, is related to the Fibonacci sequence, which deserves its own discussion.

Of course, the Fibonacci numbers are the well-known sequence

0  1  1  2  3  5  8  13  21  34,

where each consecutive term is the sum of the previous two. Although the sequence was first described by the Indians, Fibonacci is credited with introducing it to western mathematics in his rabbit problem we encountered in class. The numbers of the sequence are intimately related to the Golden Ratio, an irrational number approximately equal to 1.6180339887.... The ratio between consecutive terms of the Fibonacci sequence converge to this number. Interestingly, the Golden Ratio seems to have been known to the West before the sequence was introduced by Fibonacci; it was studied by Greek geometers, and is defined by Euclid in his Elements. The Ratio has become one of the most pervasive numbers, entering into many fields of study across the ages. In The Golden Ratio: The Story of Phi, The World's Most Astonishing Number, Mario Livio writes

"Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics." (p. 6)

After describing the rabbit population problem associated with the sequence, Fibonacci continues in Liber Abaci with a section containing a discussion of approximating irrational numbers both algebraically and geometrically, and includes proofs of various propositions in Euclidean geometry.

Fibonacci's writings in mathematics have made a lasting impression. The most significant of these contributions, in my opinion, is his introduction of the Hindu-Arabic number system. His book was written in the beginning of the 13th century, and up until that point the cumbersome Roman numeral system was the most prevalent system in the West. To this day, we (every advanced, "Westernized" society, including Asia to a degree) use the Hindu-Arabic system and variations of the methods suggested by Fibonacci. It seems likely that, if Fibonacci had not written about the new Hindu-Arabic numerals, we would have had a more efficient system than the Roman numerals by now. However, the introduction of the new system by Fibonacci has undoubtedly had an effect on the efficiency of computation. If our in-class activity working with Roman numerals was any indication of how difficult computation was before the new system, it is clear that any field, whether it is business or physics, would be much more difficult than it is today if it were not for the Hindu-Arabic numerals. Thanks Fibonacci!

For my daily work, see the embedded PDF. The original proof was found here. If the embedded PDF is too small to read clearly, the link to my original file is here.

So, now, we have justification for how this formula works, which always helps in solidifying our understanding of a mathematical concept. Even after working through the derivation of the equation, it's still amazing that it produces integer values, given that exactly no part of it (except n) is even rational.