Fibonacci
INTRODUCTION
The Fibonacci Series is a sequence of numbers pioneered by Leonardo of Pisa in 1202. Fibonacci series is a deceptively simple series, but its results and purposes are almost limitless. It has captivated and puzzled mathematicians for about 700 years, and almost everyone who has worked with it has added a new perspective, a remarkable contribution to the Fibonacci puzzle, a new piece of information about the series and how it works. Fibonacci mathematics is a continuously expanding branch of number theory, with more and more people being drawn into the complex details of Fibonacci’s prestige.
DEFINITION
Fibonacci Series, in mathematics, are series of numbers in which each member is the sum of the two preceding numbers. For example, a series beginning 0, 1 … continues as 1, 2, 3, 5, 8, 13, 21, and so forth. The series was discovered by the Italian mathematician Leonardo Fibonacci (circa 1170-c. 1240), also called Leonardo of Pisa. Fibonacci numbers have many interesting properties and are widely used in mathematics. Natural patterns, such as the spiral growth of leaves on some trees, often exhibit the Fibonacci series.
THE FORMULA
The Fibonacci numbers are the sequence of numbers defined by the linear recurrence equation
with . As a result of the definition , it is conventional to define .
The Fibonacci numbers for , 2, … are 1, 1, 2, 3, 5, 8, 13, 21 …
Fibonacci numbers can be viewed as a particular case of the Fibonacci polynomials with .
Fibonacci numbers are implemented in Mathematica as Fibonacci[n].
THE MAN BEHIND FIBONACCI SERIES
Leonardo Fibonacci or Leonardo of Pisa (1170?-1240?) was an Italian mathematician, who compiled and supplemented the mathematical knowledge of classical, Arabic, and Indian cultures, and who made contributions to the mathematical fields of algebra and number theory. Fibonacci was born in Pisa, Italy, a commercial city, where he learned the basics of business calculation. When Fibonacci was about 20, he went to Algeria, where he began to learn Indian numerals and Arabic calculating methods, knowledge he supplemented during more extensive travels. Fibonacci used this experience to improve on the commercial computing techniques he knew and to extend the work of classical mathematical writers, such as the Greek mathematicians Diophantus and Euclid.
Few works by Fibonacci still exist; he wrote on number theory, practical problems of business mathematics and surveying, advanced problems in algebra, and recreational mathematics. His writings on recreational mathematics, which were often posed as story problems, became classic mental challenges as early as the 13th century. Such problems often involved the summation of recurrent series, such as the Fibonacci series (kn = kn-1 + kn-2—for example, 1, 2, 3, 5, 8, 13,. . .), which he discovered. Each term of this series is called a Fibonacci number—the sum of the two numbers preceding it in the series. Fibonacci solved the problem of calculating the value for any entry. He was awarded a yearly salary by the Republic of Pisa in 1240, indicating the importance accorded to his work and also, possibly, public service to the city’s administration.
Ancient civilizations wrote out algebraic expressions using only occasional abbreviations. These longhand notations were cumbersome. The exponent x6, for example, required notation equivalent to x · x · x · x · x · x. By medieval times Islamic mathematicians were able to talk about arbitrarily high powers of the unknown variable x, and work out the basic algebra of polynomials (although they did not yet use modern symbolism). This included the ability to multiply, divide, and find square roots of polynomials as well as knowledge of the binomial theorem, which describes how to raise a binomial to an arbitrarily high power. Persian mathematician, astronomer, and poet Omar Khayyam showed how to express roots of cubic equations using line segments obtained by intersecting conic sections, but he could not find a formula for the roots. A Latin translation of al-Khwārizmī’s algebra text appeared in the 12th century. In the early 13th century, the great Italian mathematician Leonardo Fibonacci achieved a close approximation to the solution of the specific cubic equation x3 + 2×2 + cx = d. Because Fibonacci had traveled in Islamic lands, he probably used an Arabic method of successive approximations to reach his solution.
AS EXAMPLE OF SEQUENCE
A sequence is represented as a1, a2…, an, …. The as are numbers or quantities, distinct or not; a1 is the first term, a 2 the second term, and so on. If the expression has a last term, the sequence is finite; otherwise, it is infinite. A sequence is established or defined only if a rule is given that determines the nth term for every positive integer n; this rule may be given as a formula for the nth term. For example, all the positive integers, in natural order, form an infinite sequence; this sequence is defined by the formula an=n. The formula an = n2 determines the sequence 1, 4, 9, 16, …. The rule of starting with 0, 1, then letting each term be the sum of the two preceding terms determines the sequence 0, 1, 1, 2, 3, 5, 8, 13, …; this is known as the Fibonacci sequence.
FIBONACCI SEQUENCE
Fibonacci is perhaps best known for a simple series of numbers, introduced in Liber abaci and later named the Fibonacci numbers in his honour.
The series begins with 0 and 1. After that, use the simple rule:
Add the last two numbers to get the next.
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,…
You might ask where this came from? In Fibonacci’s day, mathematical competitions and challenges were common. For example, in 1225 Fibonacci took part in a tournament at Pisa ordered by the emperor himself, Frederick II.
It was in just this type of competition that the following problem arose:
Beginning with a single pair of rabbits, if every month each productive pair bears a new pair, which becomes productive when they are 1 month old, how many rabbits will there be after n months?
ANSWER TO RABBIT PROBLEM
Imagine that there are xn pairs of rabbits after n months. The number of pairs in month n+1 will be xn (in this problem, rabbits never die) plus the number of new pairs born. But new pairs are only born to pairs at least 1 month old, so there will be xn-1 new pairs.
xn+1 = xn + xn-1
Which is simply the rule for generating the Fibonacci numbers.
FIBONACCI IN NATURE
The rabbit breeding problem that caused Fibonacci to write about the sequence in Liber abaci may be unrealistic but the Fibonacci numbers really do appear in nature. For example, some plants branch in such a way that they always have a Fibonacci number of growing points. Flowers often have a Fibonacci number of petals, daisies can have 34, 55 or even as many as 89 petals!
Finally, next time you look at a sunflower, take the trouble to look at the arrangement of the seeds. They appear to be spiralling outwards both to the left and the right. There are a Fibonacci number of spirals! It seems that this arrangement keeps the seeds uniformly packed no matter how large the seed head.
Nature uses spirals to prevent overcrowding.
FIBONACCI IN MATHS
The Fibonacci numbers are studied as part of number theory and have applications in the counting of mathematical objects such as sets, permutations and sequences and to computer science.
OTHER EXAMPLES
The Fibonacci numbers , are squareful for , 12, 18, 24, 25, 30, 36, 42, 48, 50, 54, 56, 60, 66, …, 372, 375, 378, 384, … (Sloane’s A037917) and squarefree for , 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, … (Sloane’s A037918). and for all , and there is at least one such that . No squareful Fibonacci numbers are known with prime.
The ratios of successive Fibonacci numbers approaches the golden ratio as approaches infinity, as first proved by Scottish mathematician Robert Simson in 1753 (Wells 1986, p. 62). The ratios of alternate Fibonacci numbers are given by the convergents to , where is the golden ratio, and are said to measure the fraction of a turn between successive leaves on the stalk of a plant (phyllotaxis): 1/2 for elm and linden, 1/3 for beech and hazel, 2/5 for oak and apple, 3/8 for poplar and rose, 5/13 for willow and almond, etc. (Coxeter 1969, Ball and Coxeter 1987). The Fibonacci numbers are sometimes called pine cone numbers (Pappas 1989, p. 224). The role of the Fibonacci numbers in botany is sometimes called Ludwig’s law (Szymkiewicz 1928; Wells 1986, p. 66; Steinhaus 1999, p. 299).
FIBONACCI IN MOVIE(S)
A scrambled version 13, 3, 2, 21, 1, 1, 8, 5 (Sloane’s A117540) of the first eight Fibonacci numbers appear as one of the clues left by murdered museum curator Jacque Saunière in D. Brown’s novel The Da Vinci Code (Brown 2003, pp. 43, 60-61, and 189-192). In the Season 1 episode “Sabotage” (2005) of the television crime drama NUMB3RS, math genius Charlie Eppes mentions that the Fibonacci numbers are found in the structure of crystals and the spiral of galaxies and a nautilus shell.
Leonardo Fibonacci
Italian mathematician Leonardo Fibonacci made advances in number theory and algebra. He is especially known for his work on series of numbers, including the Fibonacci series. Each number in a Fibonacci series is equal to the sum of the two numbers that came before it.
Corbis
REFERENCES
“Fibonacci Series.” Microsoft® Encarta® 2006 [DVD]. Redmond, WA: Microsoft Corporation, 2005.
“Algebra.” Microsoft® Encarta® 2006 [DVD]. Redmond, WA: Microsoft Corporation, 2005.
Hevly, Bruce W. “Leonardo Fibonacci.” Microsoft® Encarta® 2006 [DVD]. Redmond, WA: Microsoft Corporation, 2005.
Knott, R., Quinney, D. A., PASS Maths. “The life and numbers of Fibonacci.” 7 Dec. 2006. <http://pass.maths.org.uk/issue3/fibonacci/index.html>
Corbis. “Leonardo Fibonacci.” Microsoft® Encarta® 2006 [DVD]. Redmond, WA: Microsoft Corporation, 2005.
Wolfram MathWorld. “Fibonacci Number.” 8 Dec. 2006. http://mathworld.wolfram.com/FibonacciNumber.html
