The second section of Liber abaci contains a large collection of problems aimed at merchants. They relate to the price of goods, how to calculate profit on transactions, how to convert between the various currencies in use in Mediterranean countries, and problems which had originated in China.
A problem in the third section of Liber abaci led to the introduction of the Fibonacci numbers and the Fibonacci sequence for which Fibonacci is best remembered today:
A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?
The resulting sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... (Fibonacci omitted the first term in Liber abaci). This sequence, in which each number is the sum of the two preceding numbers, has proved extremely fruitful and appears in many different areas of mathematics and science. The Fibonacci Quarterly is a modern journal devoted to studying mathematics related to this sequence.
Many other problems are given in this third section, including these types, and many many more:
A spider climbs so many feet up a wall each day and slips back a fixed number each night, how many days does it take him to climb the wall.
A hound whose speed increases arithmetically chases a hare whose speed also increases arithmetically, how far do they travel before the hound catches the hare.
Calculate the amount of money two people have after a certain amount changes hands and the proportional increase and decrease are given.
There are also problems involving perfect numbers, problems involving the Chinese remainder theorem and problems involving summing arithmetic and geometric series.
Fibonacci treats numbers such as √10 in the fourth section, both with rational approximations and with geometric constructions.
A second edition of Liber abaci was produced by Fibonacci in 1228 with a preface, typical of so many second editions of books, stating that:
... new material has been added [to the book] from which superfluous had been removed...
Another of Fibonacci's books is Practica geometriae written in 1220 which is dedicated to Dominicus Hispanus whom we mentioned above. It contains a large collection of geometry problems arranged into eight chapters with theorems based on Euclid's Elements and Euclid's On Divisions. In addition to geometrical theorems with precise proofs, the book includes practical information for surveyors, including a chapter on how to calculate the height of tall objects using similar triangles. The final chapter presents what Fibonacci called geometrical subtleties [1]:
Among those included is the calculation of the sides of the pentagon and the decagon from the diameter of circumscribed and inscribed circles; the inverse calculation is also given, as well as that of the sides from the surfaces. ... to complete the section on equilateral triangles, a rectangle and a square are inscribed in such a triangle and their sides are algebraically calculated ...
In Flos Fibonacci gives an accurate approximation to a root of 10x + 2x2 + x3 = 20, one of the problems that he was challenged to solve by Johannes of Palermo. This problem was not made up by Johannes of Palermo, rather he took it from Omar Khayyam's algebra book where it is solved by means of the intersection of a circle and a hyperbola. Fibonacci proves that the root of the equation is neither an integer nor a fraction, nor the square root of a fraction. He then continues:
And because it was not possible to solve this equation in any other of the above ways, I worked to reduce the solution to an approximation.
Without explaining his methods, Fibonacci then gives the approximate solution in sexagesimal notation as 1.22.7.42.33.4.40 (this is written to base 60, so it is 1 + 22/60 + 7/602 + 42/603 + ...). This converts to the decimal 1.3688081075 which is correct to nine decimal places, a remarkable achievement.
Liber quadratorum, written in 1225, is Fibonacci's most impressive piece of work, although not the work for which he is most famous. The book's name means the book of squares and it is a number theory book which, among other things, examines methods to find Pythogorean triples. Fibonacci first notes that square numbers can be constructed as sums of odd numbers, essentially describing an inductive construction using the formula n2 + (2n+1) = (n+1)2. Fibonacci writes:
I thought about the origin of all square numbers and discovered that they arose from the regular ascent of odd numbers. For unity is a square and from it is produced the first square, namely 1; adding 3 to this makes the second square, namely 4, whose root is 2; if to this sum is added a third odd number, namely 5, the third square will be produced, namely 9, whose root is 3; and so the sequence and series of square numbers always rise through the regular addition of odd numbers.
To construct the Pythogorean triples, Fibonacci proceeds as follows:
Thus when I wish to find two square numbers whose addition produces a square number, I take any odd square number as one of the two square numbers and I find the other square number by the addition of all the odd numbers from unity up to but excluding the odd square number. For example, I take 9 as one of the two squares mentioned; the remaining square will be obtained by the addition of all the odd numbers below 9, namely 1, 3, 5, 7, whose sum is 16, a square number, which when added to 9 gives 25, a square number.
Fibonacci also proves many interesting number theory results such as:
there is no x, y such that x2 + y2 and x2  y2 are both squares.
and x4  y4 cannot be a square.
He defined the concept of a congruum, a number of the form ab(a + b)(a  b), if a + b is even, and 4 times this if a + b is odd. Fibonacci proved that a congruum must be divisible by 24 and he also showed that for x, c such that x2 + c and x2  c are both squares, then c is a congruum. He also proved that a square cannot be a congruum.
As stated in [2]:
... the Liber quadratorum alone ranks Fibonacci as the major contributor to number theory between Diophantus and the 17thcentury French mathematician Pierre de Fermat.
Fibonacci's influence was more limited than one might have hoped and apart from his role in spreading the use of the HinduArabic numerals and his rabbit problem, Fibonacci's contribution to mathematics has been largely overlooked. As explained in [1]:
Direct influence was exerted only by those portions of the "Liber abaci" and of the "Practica" that served to introduce IndianArabic numerals and methods and contributed to the mastering of the problems of daily life. Here Fibonacci became the teacher of the masters of computation and of the surveyors, as one learns from the "Summa" of Luca Pacioli ... Fibonacci was also the teacher of the "Cossists", who took their name from the word 'causa' which was first used in the West by Fibonacci in place of 'res' or 'radix'. His alphabetic designation for the general number or coefficient was first improved by Vičte ...
Fibonacci's work in number theory was almost wholly ignored and virtually unknown during the Middle ages. Three hundred years later we find the same results appearing in the work of Maurolico.
The portrait above is from a modern engraving and is believed to not be based on authentic sources.
Article by: J J O'Connor and E F Robertson
P/S: Sekarang Fibonacci teori banyak digunakan oleh trader bagi meramal pergerakan market...


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