Fibonacci biography Version for printing Leonardo Pisano is better known by his nickname Fibonacci. He was the son of Guilielmo and a member of the Bonacci family. Fibonacci himself sometimes used the name Bigollo, which may mean good-for-nothing or a traveller. As stated in [ 1 ]:- Did his countrymen wish to express by this epithet their disdain for a man who concerned himself with questions of no practical value, or does the word in the Tuscan dialect mean a much-travelled man, which he was? Fibonacci was born in Italy but was educated in North Africa where his father, Guilielmo, held a diplomatic post. His father's job was to represent the merchants of the Republic of Pisa who were trading in Bugia, later called Bougie and now called Bejaia. Bejaia is a Mediterranean port in northeastern Algeria. The town lies at the mouth of the Wadi Soummam near Mount Gouraya and Cape Carbon. Fibonacci was taught mathematics in Bugia and travelled widely with his father and recognised the enormous advantages of the mathematical systems used in the countries they visited.

Fibonacci writes in his famous book Liber abaci (1202):- When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians' nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it, for whatever was studied by the art in Egypt, Syria, Greece, Sicily and Provence, in all its various forms. Fibonacci ended his travels around the year 1200 and at that time he returned to Pisa. There he wrote a number of important texts which played an important role in reviving ancient mathematical skills and he made significant contributions of his own. Fibonacci lived in the days before printing, so his books were hand written and the only way to have a copy of one of his books was to have another hand-written copy made.

Of his books we still have copies of Liber abaci (1202), Practica geometriae (1220), Flos (1225), and Liber quadratorum. Given that relatively few hand-made copies would ever have been produced, we are fortunate to have access to his writing in these works. However, we know that he wrote some other texts which, unfortunately, are lost. His book on commercial arithmetic Di minor guisa is lost as is his commentary on Book X of Euclid 's Elements which contained a numerical treatment of irrational numbers which Euclid had approached from a geometric point of view. One might have thought that at a time when Europe was little interested in scholarship, Fibonacci would have been largely ignored. This, however, is not so and widespread interest in his work undoubtedly contributed strongly to his importance. Fibonacci was a contemporary of Jordanus but he was a far more sophisticated mathematician and his achievements were clearly recognised, although it was the practical applications rather than the abstract theorems that made him famous to his contemporaries. The Holy Roman emperor was Frederick II.

He had been crowned king of Germany in 1212 and then crowned Holy Roman emperor by the Pope in St Peter's Church in Rome in November 1220. Frederick II supported Pisa in its conflicts with Genoa at sea and with Lucca and Florence on land, and he spent the years up to 1227 consolidating his power in Italy. State control was introduced on trade and manufacture, and civil servants to oversee this monopoly were trained at the University of Naples which Frederick founded for this purpose in 1224. Frederick became aware of Fibonacci's work through the scholars at his court who had corresponded with Fibonacci since his return to Pisa around 1200. These scholars included Michael Scotus who was the court astrologer, Theodorus Physicus the court philosopher and Dominicus Hispanus who suggested to Frederick that he meet Fibonacci when Frederick's court met in Pisa around 1225. Johannes of Palermo, another member of Frederick II's court, presented a number of problems as challenges to the great mathematician Fibonacci. Three of these problems were solved by Fibonacci and he gives solutions in Flos which he sent to Frederick II.

We give some details of one of these problems below. After 1228 there is only one known document which refers to Fibonacci. This is a decree made by the Republic of Pisa in 1240 in which a salary is awarded to:- ... the serious and learned Master Leonardo Bigollo .... This salary was given to Fibonacci in recognition for the services that he had given to the city, advising on matters of accounting and teaching the citizens. Liber abaci, published in 1202 after Fibonacci's return to Italy, was dedicated to Scotus. The book was based on the arithmetic and algebra that Fibonacci had accumulated during his travels. The book, which went on to be widely copied and imitated, introduced the Hindu-Arabic place-valued decimal system and the use of Arabic numerals into Europe. Indeed, although mainly a book about the use of Arab numerals, which became known as algorism, simultaneous linear equations are also studied in this work. Certainly many of the problems that Fibonacci considers in Liber abaci were similar to those appearing in Arab sources. 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 17th-century 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 Hindu-Arabic 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 Indian-Arabic 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 Viète ... 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 Click on this link to see a list of the Glossary entries for this page Fibonacci - Medieval Mathematics - The Story of Mathematics The Story of Mathematics MEDIEVAL MATHEMATICS - FIBONACCI Leonardo of Pisa (Fibonacci) (c.1170-1250) The 13th Century Italian Leonardo of Pisa, better known by his nickname Fibonacci, was perhaps the most talented Western mathematician of the Middle Ages. Little is known of his life except that he was the son of a customs offical and, as a child, he travelled around North Africa with his father, where he learned about Arabic mathematics. On his return to Italy, he helped to disseminate this knowledge throughout Europe, thus setting in motion a rejuvenation in European mathematics, which had lain largely dormant for centuries during the Dark Ages.

In particular, in 1202, he wrote a hugely influential book called �Liber Abaci� ("Book of Calculation"), in which he promoted the use of the Hindu-Arabic numeral system, describing its many benefits for merchants and mathematicians alike over the clumsy system of Roman numerals then in use in Europe. Despite its obvious advantages, uptake of the system in Europe was slow (this was after all during the time of the Crusades against Islam, a time in which anything Arabic was viewed with great suspicion), and Arabic numerals were even banned in the city of Florence in 1299 on the pretext that they were easier to falsify than Roman numerals. However, common sense eventually prevailed and the new system was adopted throughout Europe by the 15th century, making the Roman system obsolete. The horizontal bar notation for fractions was also first used in this work (although following the Arabic practice of placing the fraction to the left of the integer). The discovery of the famous Fibonacci sequence Fibonacci is best known, though, for his introduction into Europe of a particular number sequence, which has since become known as Fibonacci Numbers or the Fibonacci Sequence.

He discovered the sequence - the first recursive number sequence known in Europe - while considering a practical problem in the �Liber Abaci� involving the growth of a hypothetical population of rabbits based on idealized assumptions. He noted that, after each monthly generation, the number of pairs of rabbits increased from 1 to 2 to 3 to 5 to 8 to 13, etc, and identified how the sequence progressed by adding the previous two terms (in mathematical terms, Fn = Fn-1 + Fn-2), a sequence which could in theory extend indefinitely. The sequence, which had actually been known to Indian mathematicians since the 6th Century, has many interesting mathematical properties, and many of the implications and relationships of the sequence were not discovered until several centuries after Fibonacci's death.

For instance, the sequence regenerates itself in some surprising ways: every third F-number is divisible by 2 (F3 = 2), every fourth F-number is divisible by 3 (F4 = 3), every fifth F-number is divisible by 5 (F5 = 5), every sixth F-number is divisible by 8 (F6 = 8), every seventh F-number is divisible by 13 (F7 = 13), etc. The numbers of the sequence has also been found to be ubiquitous in nature: among other things, many species of flowering plants have numbers of petals in the Fibonacci Sequence; the spiral arrangements of pineapples occur in 5s and 8s, those of pinecones in 8s and 13s, and the seeds of sunflower heads in 21s, 34s, 55s or even higher terms in the sequence; etc.

The Golden Ratio φ can be derived from the Fibonacci Sequence In the 1750s, Robert Simson noted that the ratio of each term in the Fibonacci Sequence to the previous term approaches, with ever greater accuracy the higher the terms, a ratio of approximately 1 : 1.6180339887 (it is actually an irrational number equal to (1 + √5)⁄2 which has since been calculated to thousands of decimal places). This value is referred to as the Golden Ratio, also known as the Golden Mean, Golden Section, Divine Proportion, etc, and is usually denoted by the Greek letter phi φ (or sometimes the capital letter Phi Φ). Essentially, two quantities are in the Golden Ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The Golden Ratio itself has many unique properties, such as 1⁄φ = φ - 1 (0.618...) and φ2 = φ + 1 (2.618...), and there are countless examples of it to be found both in nature and in the human world.

A rectangle with sides in the ratio of 1 : φ is known as a Golden Rectangle, and many artists and architects throughout history (dating back to ancient Egypt and Greece , but particularly popular in the Renaissance art of Leonardo da Vinci and his contemporaries) have proportioned their works approximately using the Golden Ratio and Golden Rectangles, which are widely considered to be innately aesthetically pleasing. An arc connecting opposite points of ever smaller nested Golden Rectangles forms a logarithmic spiral, known as a Golden Spiral. The Golden Ratio and Golden Spiral can also be found in a surprising number of instances in Nature, from shells to flowers to animal horns to human bodies to storm systems to complete galaxies. It should be remembered, though, that the Fibonacci Sequence was actually only a very minor element in �Liber Abaci� - indeed, the sequence only received Fibonacci's name in 1877 when Eduouard Lucas decided to pay tribute to him by naming the series after him - and that Fibonacci himself was not responsible for identifying any of the interesting mathematical properties of the sequence, its relationship to the Golden Mean and Golden Rectangles and Spirals, etc.

Fibonacci introduced lattice multiplication to Europe However, the book's influence on medieval mathematics is undeniable, and it does also include discussions of a number of other mathematical problems such as the Chinese Remainder Theorem, perfect numbers and prime numbers, formulas for arithmetic series and for square pyramidal numbers, Euclidean geometric proofs, and a study of simultaneous linear equations along the lines of Diophantus and Al-Karaji. He also described the lattice (or sieve) multiplication method of multiplying large numbers, a method - originally pioneered by Islamic mathematicians like Al-Khwarizmi - algorithmically equivalent to long multiplication. Neither was �Liber Abaci� Fibonacci�s only book, although it was his most important one. His �Liber Quadratorum� (�The Book of Squares�), for example, is a book on algebra, published in 1225 in which appears a statement of what is now called Fibonacci's identity - sometimes also known as Brahmagupta �s identity after the much earlier Indian mathematician who also came to the same conclusions - that the product of two sums of two squares is itself a sum of two squares e.g.

(12 + 42)(22 + 72) = 262 + 152 = 302 + 12. Who was Fibonacci? Who was Fibonacci? References to Fibonacci's Life and Times Who was Fibonacci? The "greatest European mathematician of the middle ages", his full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa ,Italy (see Pisa on Google Earth ), the city with the famous Leaning Tower, about 1175 AD. Pisa was an important commercial town in its day and had links with many Mediterranean ports. Leonardo's father, Guglielmo Bonacci, was a kind of customs officer in the present-day Algerian town of Béjaïa, (see Bejaia on Google Earth ) formerly known as Bugia or Bougie, where wax candles were exported to France. They are still called "bougies" in French. So Leonardo grew up with a North African education under the Moors and later travelled extensively around the Mediterranean coast. He would have met with many merchants and learned of their systems of doing arithmetic.

He soon realised the many advantages of the "Hindu-Arabic" system over all the others. D E Smith points out that another famous Italian - St Francis of Assisi (a nearby Italian town) - was also alive at the same time as Fibonacci: St Francis was born about 1182 (after Fibonacci's around 1175) and died in 1226 (before Fibonacci's death commonly assumed to be around 1250). By the way, don't confuse Leonardo of Pisa with Leonardo da Vinci ! Vinci was just a few miles from Pisa on the way to Florence, but Leonardo da Vinci was born in Vinci in 1452, about 200 years after the death of Leonardo of Pisa (Fibonacci). [The portrait here is a link to the University of St Andrew's site which has more on Fibonacci himself, his life and works.] His names Leonardo of Pisa is now known as Fibonacci [pronounced fib-on-arch-ee] short for filius Bonacci.

There are a couple of explanations for the meaning of Fibonacci: Fibonacci is a shortening of the Latin "filius Bonacci", used in the title of his book Libar Abaci (of which mmore later), which means "the son of Bonaccio". His father's name was Guglielmo Bonaccio. Fi'-Bonacci is like the English names of Robin-son and John-son. But (in Italian) Bonacci is also the plural of Bonaccio; therefore, two early writers on Fibonacci (Boncompagni and Milanesi) regard Bonacci as his family name (as in "the Smiths" for the family of John Smith). Fibonacci himself wrote both "Bonacci" and "Bonaccii" as well as "Bonacij"; the uncertainty in the spelling is partly to be ascribed to this mixture of spoken Italian and written Latin, common at that time. However he did not use the word "Fibonacci".

This seems to have been a nickname probably originating in the works of Guillaume Libri in 1838, accordigng to L E Sigler's in his Introduction to Leonardo Pisano's Book of Squares (see Fibonacci's Mathematical Books below). Others think Bonacci may be a kind of nick-name meaning "lucky son" (literally, "son of good fortune"). Other names He is perhaps more correctly called Leonardo of Pisa or, using a latinisation of his name, Leonardo Pisano. Occasionally he also wrote Leonardo Bigollo since, in Tuscany, bigollo means a traveller. We shall just call him Fibonacci as do most modern authors, but if you are looking him up in older books, be prepared to see any of the above variations of his name. [With thanks to Prof. Claudio Giomini of Rome for help on the Latin and Italian names in this section.] References D E Smith's History of Mathematics Volume 1, (Dover, 1958 - a reprint of the orignal version from 1923) gives a complete list of other books that he wrote and is a fuller reference on Fibonacci's life and works.

There is another brief biography of Fibonacci which is part of Karen Hunger Pashall's (Virginia University) The art of Algebra from from al-Khwarizmi to Viéte: A Study in the Natural Selection of Ideas if you want to read more about the history of mathematics. Eight Hundred Years Young by A F Horadam (University of New England) in The Australian Mathematics Teacher Vol 31, 1985, pages 123-134, is an interesting and readable article on Fibonacci, his names and origins as well as his mathematical works. He refers to and expands upon the following article... The Autobiography of Leonardo Pisano R E Grimm, in Fibonacci Quarterly vol 11, 1973, pages 99-104. Leonard of Pisa and the New Mathematics of the Middle Ages by J and F Gies, Thomas Y Crowell publishers, 1969, 127 pages, is another book with much on the background to Fibonacci's life and work.

Della vita e delle opere di Leonardo Pisano Baldassarre Boncompagni, Rome, 1854 is the only complete printed version of Fibonacci's 1228 edition of Liber Abaci. The the Math Forum 's archives of the History of Mathematics discussion group contain a useful discussion on some of the controversial topics of Fibonacci's names and life (February 1999). Use its next>> link to follow the thread of the discussion through its 6 emailed contributions. It talks about the uncertainlty of his birth and death dates and his names. It seems that Fibonacci never referred to himself as "Fibonacci" but this was a nick-name given to him by later writers.

Fibonacci's Mathematical Contributions Introducing the Decimal Number system into Europe He was one of the first people to introduce the Hindu-Arabic number system into Europe - the positional system we use today - based on ten digits with its decimal point and a symbol for zero: 1 2 3 4 5 6 7 8 9 0 His book on how to do arithmetic in the decimal system, called Liber abbaci (meaning Book of the Abacus or Book of Calculating) completed in 1202 persuaded many European mathematicians of his day to use this "new" system.

The book describes (in Latin) the rules we all now learn at elementary school for adding numbers, subtracting, multiplying and dividing, together with many problems to illustrate the methods: 1 7 4 + 1 7 4 - 1 7 4 x 1 7 4 ÷ 28 2 8 2 8 2 8 is ----- ----- ------- 2 0 2 1 4 6 3 4 8 0 + 6 remainder 6 ----- ----- 1 3 9 2 ------- 4 8 7 2 ------- Let's first of all look at the Roman number system still in use in Europe at that time (1200) and see how awkward it was for arithmetic. Roman Numerals The method in use in Europe until then used the Roman numerals: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500 and M = 1000 You can still see them used on foundation stones of old buildings and on some clocks.

The Additive rule The simplest system would be merely to use the letters for the values as in the table above, and add the values for each letter used. For instance, 13 could be written as XIII or perhaps IIIX or even IIXI. This occurs in the Roman language of Latin where 23 is spoken as tres et viginti which translates as three and twenty. You may remember the nursery rhyme Sing a Song of Sixpence which begins Sing a song of sixpence A pocket full of rye Four and twenty blackbirds Baked in a pie... Above 100, the Latin words use the same order as we do in English, so that whereas 35 is quinque et triginta (5 and 30), 235 is ducenti triginta quinque (two hundred thirty five).

In this simple system, using addition only, 99 would be 90+9 or, using only the numbers above, 50+10+10+10 + 5+1+1+1+1 which translates to LXXXXVIIII and by the same method 1998 would be written by the Romans as MDCCCCLXXXXVIII. But some numbers are long and it is this is where, if we agree to let the order of letters matter we can also use subtraction. The subtractive rule The Roman language (Latin) also uses a subtraction principle so that whereas 20 is viginti 19 is "1 from 20" or undeviginti. We have it in English when we say the time is "10 to 7" which is not the same as "7 10". The first means 10 minutes before ( or subtracted from) 7 0'clock, whereas the second means 10 minutes added to (or after) 7 o'clock. This is also reflected in Roman numerals. This abbreviation makes the order of letters important.

So if a smaller value came before the next larger one, it was subtracted and if it came after, it was added. For example, XI means 10+1=11 (since the smaller one comes after the larger ten) but IX means 1 less than 10 or 9. But 8 is still written as VIII (not IIX). The subtraction in numbers was only of a unit (1, 10 or 100) taken away from 5 of those units (5, 50 or 500 or from the next larger multiple of 10 (10, 100 or 1000). Using this method, 1998 would be written much more compactly as MCMXCVIII but this takes a little more time to interpret: 1000 + (100 less than 1000) + (10 less than 100) + 5 + 1 + 1 + 1.

Note that in the UK we use a similar system for time when 6:50 is often said as "ten to 7" as well as "6 fifty", similarly for "a quarter to 4" meaning 3:45. In the USA, 6:50 is sometimes spoken as "10 of 7". Look out for Roman numerals used as the date a film was made, often recorded on the screen which gives its censor certification or perhaps the very last image of the movie giving credits or copyright information. Arithmetic with Roman Numerals For more on Roman Numerals, see the excellent Frequently Asked Questions on Roman Numerals at Math Forum. The Decimal Positional System The system that Fibonacci introduced into Europe came from India and Arabia and used the Arabic symbols 1, 2, 3, 4, 5, 6, 7, 8, 9 with, most importantly, a symbol for zero 0. With Roman numbers, 2003 could be written as MMIII or, just as clearly, it could be written as IIIMM - the order does not matter since the values of the letters are added to make the number in the original (unabbreviated) system.

With the abbreviated system of IX meaning 9, then the order did matter but it seems this sytem was not often used in Roman times. In the "new system", the order does matter always since 23 is quite a different number to 32. Also, since the position of each digit is important, then we may need a zero to get the digits into their correct places (columns) eg 2003 which has no tens and no hundreds. (The Roman system would have just omitted the values not used so had no need of "zero".) This decimal positional system, as we call it, uses the ten symbols of Arabic origin and the "methods" used by Indian Hindu mathematicians many years before they were imported into Europe. It has been commented that in India, the concept of nothing is important in its early religion and philosophy and so it was much more natural to have a symbol for it than for the Latin (Roman) and Greek systems.

"Algorithm" Earlier the Persian author Abu ‘Abd Allah, Mohammed ibn Musa al-Khwarizmi (usually abbreviated to Al-Khwarizmi had written a book which included the rules of arithmetic for the decimal number system we now use, called Kitab al jabr wa‘l-muqabala (Rules of restoring and equating) dating from about 825 AD. D E Knuth (in the errata for the second edition and third edition of his "Fundamental Algorithms") gives the full name above and says it can be translated as Father of Abdullah, Mohammed, son of Moses, native of Khwarizm. He was an astromomer to the caliph at Baghdad (now in Iraq). Al-Khowârizmî is the region south and to the east of the Aral Sea around the town now called Khiva (or Urgench) on the Amu Darya river. It was part of the Silk Route, a major trading pathway between the East and Europe. In 1200 it was in Persia but today is in Uzbekistan , part of the former USSR, north of Iran, which gained its independence in 1991.

Prof Don Knuth has a picture of a postage stamp issued by the USSR in 1983 to commemorate al-Khowârizmî 1200 year anniversary of his probable birth date. From the title of this book Kitab al jabr w'al-muqabala we derive our modern word algebra. The Persian author's name is commemorated in the word algorithm. It has changed over the years from an original European pronunciation and latinisation of algorism. Algorithms were known of before Al-Khowârizmî's writings, (for example, Euclid's Elements is full of algorithms for geometry, including one to find the greatest common divisor of two numbers called Euclid's algorithm today). The USA Library of Congress has a list of citations of Al-Khowârizmî and his works.

Our modern word "algorithm" does not just apply to the rules of arithmetic but means any precise set of instructions for performing a computation whether this be a method followed by humans, for example: a cooking recipe; a car manual page for example, on how to remove the gear-box; a medical procedure such as removing your appendix; a calculation by human computors : two examples are: William Shanks who computed the value of pi to 707 decimal places by hand last century over about 20 years up to 1873 - but he was wrong at the 526-th place when it was checked by desk calculators in 1944! Earlier Johann Dase had computed pi correctly to 205 decimal places in 1844 when aged 20 but this was done completely in his head just writing the number down after working on it for two months!! or mechanically by machines (such as placing chips and components at correct places on a circuit board to go inside your TV) or automatically by electronic computers which store the instructions as well as data to work on.

See D E Knuth, The Art of Computer Programming Volume 1: Fundamental Algorithms (now in its Third Edition, 1997)pages 1-2. There is an English translation of the ".. al jabr .." book: L C Karpinski Robert of Chester's Latin Translation ... of al-Khowarizmi published in New York in 1915. [Note the variation in the spelling of "Al-Khowârizmî" here - this is not unusual! Other spellings include al-Khorezmi.] Ian Stewart's The Problems of Mathematics (Oxford) 1992, ISBN: 0-19-286148-4 has a chapter on algorithms and the history of the name: chapter 21: Dixit Algorizmi.

The Fibonacci Numbers In Fibonacci's Liber Abaci book, chapter 12, he introduces the following problem (here in Sigler's translation - see below): How Many Pairs of Rabbits Are Created by One Pair in One Year A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also. He then goes on to solve and explain the solution: Because the above written pair in the first month bore, you will double it; there will be two pairs in one month. One of these, namely the first, bears in the second montth, and thus there are in the second month 3 pairs; of these in one month two are pregnant and in the third month 2 pairs of rabbits are born, and thus there are 5 pairs in the month; ... there will be 144 pairs in this [the tenth] month; to these are added again the 89 pairs that are born in the eleventh month; there will be 233 pairs in this month.

To these are still added the 144 pairs that are born in the last month; there will be 377 pairs, and this many pairs are produced from the abovewritten pair in the mentioned place at the end of the one year. You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the 1 to the 2, and the second to the third, and the third to the fourth and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the abovewritten sum of rabbits, namely 377, and thus you can in order find it for an unending number of months. beginning 1 end 377 Did Fibonacci invent this Series? Fibonacci says his book Liber Abaci (the first edition was dated 1202) that he had studied the "nine Indian figures" and their arithmetic as used in various countries around the Mediterranean and wrote about them to make their use more commonly understood in his native Italy.

So he probably merely included the "rabbit problem" from one of his contacts and did not invent either the problem or the series of numbers which now bear his name. D E Knuth adds the following in his monumental work The Art of Computer Programming: Volume 1: Fundamental Algorithms errata to second edition : Before Fibonacci wrote his work, the sequence F(n) had already been discussed by Indian scholars, who had long been interested in rhythmic patterns that are formed from one-beat and two-beat notes. The number of such rhythms having n beats altogether is F(n+1); therefore both Gospala (before 1135) and Hemachandra (c. 1150) mentioned the numbers 1, 2, 3, 5, 8, 13, 21, ... explicitly. Knuth refers to an article by P Singh in Historia Mathematica vol 12 (1985) pages 229-244.

Naming the Series It was the French mathematician Edouard Lucas (1842-1891) who gave the name Fibonacci numbers to this series and found many other important applications as well as having the series of numbers that are closely related to the Fibonacci numbers - the Lucas Numbers : 2, 1, 3, 4, 7, 11, 18, 29, 47, ... named after him. Fibonacci memorials to see in Pisa He died in the 1240's and there is now a statue commemorating him located at the Leaning Tower end of the cemetery next to the Cathedral in Pisa. [With special thanks to Nicholas Farhi , an ex-pupil of Winchester College, for the picture of the statue.] The picture of Pisa's cathedral and leaning tower is a link to more information on Pisa. Clark Kimberling, Professor of Mathematics at Evansville University, Indiana, has a Fibonacci biography page . It shows the face of another Fibonacci statue down by the Arno river off the Via Fibonacci.

Fibonacci's Mathematical Books Leonardo of Pisa wrote 5 mathematical works, 4 as books and one preserved as a letter: Fibonacci's Liber Abaci translated by L E Sigler, Springer Verlag (2002), 672 pages available for the first time in English in 2002 celebrating it's 800th anniversary, as a translation with notes of Fibonacci's Liber Abaci (The Book of Calculating) from 1202 but revised in 1228. One of the problems in this book was the problem about the rabbits in a field which introduced the series 1, 2, 3, 5, 8, ... . It was much later (around 1870) that E Lucas named this series of numbers after Fibonacci. Mathematician Leonardo Pisano Fibonacci - Life and Works Updated January 30, 2016. Background Also referred to as Leonard of Pisa, Fibonacci was an Itallian number theorist. It is believed that Leonardo Pisano Fibonacci was born in the 13th century, in 1170 (approximately) and that he died in 1250.

Fibonacci was born in Italy but obtained his education in North Africa. Very little is known about him or his family and there are no photographs or drawings of him. Much of the information about Fibonacci has been gathered by his autobiographical notes which he included in his books. However, Fibonacci is considered to be one of the most talented mathematicians for the Middle Ages. Few people realize that it was Fibonacci that gave us our decimal number system (Hindu-Arabic numbering system) which replaced the Roman Numeral system. When he was studying mathematics, he used the Hindu-Arabic (0-9) symbols instead of Roman symbols which didn't have 0's and lacked place value . In fact, when using the Roman Numeral system, an abacus was usually required. continue reading below our video What are the Seven Wonders of the World There is no doubt that Fibonacci saw the superiority of using Hindu-Arabic system over the Roman Numerals. He shows how to use our current numbering system in his book Liber abaci. The following problem was written in his book called Liber abaci: 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? It was this problem that led Fibonacci to the introduction of the Fibonacci Numbers and the Fibonacci Sequence which is what he remains famous for to this day. The sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... This sequence, shows that each number is the sum of the two preceding numbers. It is a sequence that is seen and used in many different areas of mathematics and science. The sequence is an example of a recursive sequence. The Fibonacci Sequence defines the curvature of naturally occurring spirals, such as snail shells and even the pattern of seeds in flowering plants. The Fibonacci sequence was actually given the name by a French mathematician Edouard Lucas in the 1870's. Contributions Fibonacci is famous for his contributions to number theory. In his book, Liber abaci he introduced the Hindu-Arabic place-valued decimal system and the use of Arabic numerals into Europe.

He introduced us to the bar we use in fractions, previous to this, the numerator has quotations around it. The square root notation is also a Fibonacci method. It has been said that the Fibonacci numbers are Nature's numbering system and apply to the growth of living things, including cells, petals on a flower, wheat, honeycomb, pine cones and much more. His Books: Leonardo Pisano | Italian mathematician | Britannica.com Italian mathematician Alternative Titles: Leonardo Fibonacci, Leonardo of Pisa Leonardo Pisano Leonardo Pisano, English Leonardo of Pisa, original name Leonardo Fibonacci (born c. 1170, Pisa ?—died after 1240), medieval Italian mathematician who wrote Liber abaci (1202; “Book of the Abacus”), the first European work on Indian and Arabian mathematics . Life Little is known about Leonardo’s life beyond the few facts given in his mathematical writings. During Leonardo’s boyhood his father, Guglielmo, a Pisan merchant, was appointed consul over the community of Pisan merchants in the North African port of Bugia (now Bejaïa , Alg.). Leonardo was sent to study calculation with an Arab master.

He later went to Egypt, Syria, Greece, Sicily, and Provence, where he studied different numerical systems and methods of calculation. When Leonardo’s Liber abaci first appeared, Hindu-Arabic numerals were known to only a few European intellectuals through translations of the writings of the 9th-century Arab mathematician al-Khwārizmī. The first seven chapters dealt with the notation, explaining the principle of place value, by which the position of a figure determines whether it is a unit, 10, 100, and so forth, and demonstrating the use of the numerals in arithmetical operations. The techniques were then applied to such practical problems as profit margin, barter, money changing, conversion of weights and measures, partnerships, and interest. Most of the work was devoted to speculative mathematics— proportion (represented by such popular medieval techniques as the Rule of Three and the Rule of Five, which are rule-of-thumb methods of finding proportions), the Rule of False Position (a method by which a problem is worked out by a false assumption, then corrected by proportion), extraction of roots, and the properties of numbers, concluding with some geometry and algebra.

In 1220 Leonardo produced a brief work, the Practica geometriae (“Practice of Geometry”), which included eight chapters of theorems based on Euclid’s Elements and On Divisions. The Liber abaci, which was widely copied and imitated, drew the attention of the Holy Roman emperor Frederick II . In the 1220s Leonardo was invited to appear before the emperor at Pisa, and there John of Palermo, a member of Frederick’s scientific entourage, propounded a series of problems, three of which Leonardo presented in his books. The first two belonged to a favourite Arabic type, the indeterminate, which had been developed by the 3rd-century Greek mathematician Diophantus. This was an equation with two or more unknowns for which the solution must be in rational numbers (whole numbers or common fractions).

The third problem was a third-degree equation (i.e., containing a cube), x3 + 2x2 + 10x = 20 (expressed in modern algebraic notation), which Leonardo solved by a trial-and-error method known as approximation; he arrived at the answer Britannica Stories in sexagesimal fractions (a fraction using the Babylonian number system that had a base of 60), which, when translated into modern decimals (1.3688081075), is correct to nine decimal places. Contributions to number theory For several years Leonardo corresponded with Frederick II and his scholars, exchanging problems with them. He dedicated his Liber quadratorum (1225; “Book of Square Numbers”) to Frederick. Devoted entirely to Diophantine equations of the second degree (i.e., containing squares), the Liber quadratorum is considered Leonardo’s masterpiece. It is a systematically arranged collection of theorems, many invented by the author, who used his own proofs to work out general solutions. Probably his most creative work was in congruent numbers—numbers that give the same remainder when divided by a given number.

He worked out an original solution for finding a number that, when added to or subtracted from a square number, leaves a square number. His statement that x2 + y2 and x2 - y2 could not both be squares was of great importance to the determination of the area of rational right triangles. Although the Liber abaci was more influential and broader in scope, the Liber quadratorum alone ranks Leonardo as the major contributor to number theory between Diophantus and the 17th-century French mathematician Pierre de Fermat . Numbers and Mathematics Except for his role in spreading the use of the Hindu-Arabic numerals, Leonardo’s contribution to mathematics has been largely overlooked. His name is known to modern mathematicians mainly because of the Fibonacci sequence (see below) derived from a problem in the Liber abaci: 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 number sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 (Leonardo himself omitted the first term), in which each number is the sum of the two preceding numbers, is the first recursive number sequence (in which the relation between two or more successive terms can be expressed by a formula) known in Europe. Terms in the sequence were stated in a formula by the French-born mathematician Albert Girard in 1634: un + 2 = un + 1 + un, in which u represents the term and the subscript its rank in the sequence. The mathematician Robert Simson at the University of Glasgow in 1753 noted that, as the numbers increased in magnitude, the ratio between succeeding numbers approached the number α, the golden ratio , whose value is 1.6180 . . . , or (1 + √5)/2.

In the 19th century the term Fibonacci sequence was coined by the French mathematician Edouard Lucas , and scientists began to discover such sequences in nature; for example, in the spirals of sunflower heads, in pine cones, in the regular descent (genealogy) of the male bee, in the related logarithmic (equiangular) spiral in snail shells, in the arrangement of leaf buds on a stem, and in animal horns. Famous Mathematicians - Facts & Information Logic Puzzles Money Addition Subtraction Multiplication Division Counting Problem Solving Sudoku Board Games Memory Fractions Probability Famous Mathematicians Learn about the famous mathematicians whose incredible work has helped change both the field of mathematics and the world we live in today. Enjoy a range of facts and information about brilliant mathematicians such as Pythagoras, Euclid, Pierre Fermat, Blaise Pascal, Leonhard Euler, Carl Friedrich Gauss, Alan Turing and more. Read what they did, when they did it and enjoy learning about the most famous mathematicians of all time.

Greek philosopher and mathematician Pythagoras lived around the year 500 BC and is known for his Pythagorean theorem relating to the three sides of a right angle triangle: a² + b² = c² Greek mathematician Euclid is often referred to as the ‘Father of Geometry’ for his revolutionary ideas and influential textbook called ‘Elements’ that he wrote around the year 300 BC. Archimedes of Syracuse lived around the year 250 BC and among other things, developed a method for determining the volume of objects with irregular shapes. Italian mathematician Leonardo of Pisa (better known as Fibonacci) lived between the years 1170 and 1250 and is best known today for Fibonacci numbers, the number sequence named after him. Fibonacci introduced the number sequence to Western Europe in his book ‘Liber Abaci’ after they had been described earlier by Indian mathematicians.

The Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, .... In the 17th century Galileo Galilei and Johannes Kepler made important discoveries relating to planetary motion and orbits. German mathematician Gottfried Leibniz lived between 1646 and 1716, developing important calculus concepts and mathematical notation practices. Isaac Newton discovered the laws of physics and brought together many important concepts of infinitesimal calculus. Much of the work done by Leibniz and Newton is based on theories by French philosopher Rene Descartes. As well as his many contributions to philosophy, Descartes also had a huge impact on mathematics, creating analytical geometry, developing a system that describes geometry using algebra, contributing to optics and much more. Born in France, Pierre de Fermat was an amateur mathematician who is best known for Fermat’s Last Theorem. In 1642 French mathematician Blaise Pascal invented the mechanical calculator.

Swiss mathematician Leonhard Euler was probably the most influential mathematician of the 18th century, making discoveries in graph theory and introducing many modern mathematical words and notations among other things. Born in 1777, German mathematician Carl Friedrich Gauss contributed brilliant work in geometry, statistics, number theories, algebra and much more. Bernhard Riemann was an influential German mathematician who contributed to differential geometry and analysis, paving the way for the development of general relativity by Albert Einstein. Born in 1882, Emmy Noether was a German mathematician who made important contributions to abstract algebra and theoretical physics, described by Einstein as the most important woman in the history of mathematics. Alan Turing was a British mathematician and computer scientist who cracked German ciphers (codes) in the Second World War, contributed to mathematical logic and played an important role in the development of algorithms, artificial intelligence and the modern computer. Born in 1953, British mathematician Andrew Wiles is most famous for proving Fermat’s Last Theorem.

Fibonacci Fibonacci History of Mathematics Term Paper, Rutgers, Spring 1999 The Many Names Leonardo Pisano Fibonacci was born in 1170 in Pisa [1, p. 604]. His name at birth was simply Leonardo, but in popular works today he is most commonly referred to as Fibonacci (from filio Bonacij, literally meaning son of Bonacci, but here taken as of the family Bonacci, since his father's name was not Bonacci, according to [1, p. 604]). Interestingly enough there is no proof that Fibonacci was known as such in his own time, and it has been suggested that the name Fibonacci originated with Guillame Libri [3, xv]. Fibonacci was also known by the nickname "Bigollo", which may be taken to mean loafer, and may have expressed the general lack of interest in the purely theoretical mathematics Fibonacci showed interest in. However Bigollo may also be interpreted as "well traveled", which adequately describes Fibonacci [2]. However these explanations of the nickname Bigollo have also been termed "fanciful", and having "no merit" [3, xv].

Fibonacci used the name Leonardo Pisano, which simply made reference to his origin in the city of Pisa. Now that the many names of Leonardo Pisano Bigollo Fibonacci have been set forth, from this point on I will simply refer to him as Fibonacci. Biography Fibonacci was born in 1170 to Guilielmo, a member of the Bonacci family. Guilielmo held a position as a secretary of the Republic of Pisa , in the Province of Tuscany, which at the time was part of the Holy Roman Empire, and today is in northern Italy [1, xv]. In approximately 1192 Guilielmo was posted to a trading center in the city of Bugia (now known as Bougie [1, p. 604] or as Bijaya by the Arabs [4]), which is in northeastern Algeria, and lies fairly close to Italy, across the Mediterranean Sea. Guilielmo brought Fibonacci to Bugia when he accepted this position. In Bugia Fibonacci learned a great many things - notably, from the point of view of the history of mathematics, the Indian numerals.

Fibonacci wrote of his stay in Bugia in the Liber abbaci [2] as follows: When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians' nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it, for whatever was studied by the art in Egypt, Syria, Greece, Sicily and Provence, in all its various forms. In his travels Fibonacci learned a great deal of mathematics, including the Indian numerals, and also a great many of the Greek classics which had been lost to the west, but whose Latin translations had been preserved in the east through the long endurance of the Roman empire in Constantinople [1, p607].

Fibonacci ended his travels in approximately 1200 and settled down in Pisa, where for the next twenty five years he composed a number of texts in which he did important work in number theory and the solution of algebraic equations among other important things. He also came upon the series of numbers known today as the Fibonacci numbers. Fibonacci gained recognition in the court of Emperor Frederick II. Fibonacci is believed to have died around 1250, but in any case some time after 1240; there are no records of him after this date [3]. Important Works Fibonacci's first major work was the Liber abbaci ("The Book of Calculations" [5]) written in 1202. The Liber abbaci was subsequently re-edited by Fibonacci himself in 1228. In the Liber abbaci Fibonacci presents an overview of basic arithmetic and algebra. First he discusses common finger computations and the use of Roman numerals, which were common computational methods in Europe at that time [1, p. 608]. Next he introduces the Indian numerals.

The Liber abbaci begins"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 [5]." Next he explains the Arabic rules for working with the Indian Numerals [1, p. 606]. Fractions are placed to the left of integers, and the fraction bar is used. The basic rules for these are outlined below (the examples are based on examples in the Dictionary of Scientific Biography, p. 607 and Lüneburg, pp. 68-81.) The Liber abbaci also examines root extraction, and a variety of word problems, which include many classical problems similar to those in ancient Egyptian sources [1, p. 606]. Many of these word problems lead to problems in indeterminate analysis, such as this example [1, p.

606]: x+b=2(y-7) y+b=3(x+z) z+b=4(x+y) The name indeterminate analysis refers to the fact that the set of equations is underdetermined: there are four unknowns `b,x,y,z' and only three equations, so that a unique answer is not possible, but only a relation between values of the variables. The Liber abbaci also contains many practical problems of value to merchants of the time, ranging from the calculation of interest to problems concerning currency exchange rates and profit margins. Also contained in the Liber abbaci are a variety of problems and puzzles including a famous problem on the reproduction of rabbits, which will be examined below. Fibonacci's second work is entitled the Practica geometriae, and was composed ca. 1220-1221. The Practica geometriae draws heavily on the works of the ancient Greek masters, including Euclid and Archimedes. Fibonacci also draws on the works of Plato of Tivloli (1145) [1, p. 609]. Fibonacci's discussion leads often to quadratic equations, in whose solution he shows adept skill, even taking notice of their multiple solutions.

Included in the Practica geometriae are many instructions given for the practical surveyor. Simplified instructions are given for measurement, and easily read tables are given where complex computation would have been necessary to obtain the solution. Archimedes' method of determination of ? from inscribed and circumscribed polygons is discussed. Indeterminate problems that follow from these lines of thought are also treated. The work Flos, composed in 1225, was sent to Emperor Frederick II as a response to questions set forth by Johannes of Palermo (a member of Emperor Frederick's court and a friend of Fibonacci's [3]). The first problem was the solution of the equations x2+5=y2 and x2-5=z2. The next problem is the solution of the third order equation x3+2x2+10x=20. Fibonacci showed that the solution of the equation was not a whole number, a fraction or any of the Euclidean irrational magnitudes. He does however go on to provide a rational approximation of the solution. The third and final problem is the solution of a series of indeterminate equations. In the Liber quadatorum composed in 1225 Fibonacci obtains many notable achievements in number theory.

He gives a variety of methods for finding Pythagorean triples. He next goes through a definition for a special class of numbers he calls a congruum, and then through a brilliant proof obtains the full solution of the set of indeterminate equations set forth by Johannes of Palermo. Fibonacci did not express his mathematics in the form of equations as is done today, but he did however word them in such a way as to be directly translatable into modern equations: an example from the Liber quadatorum [3, p.

4] runs: "I thought about the origin of all square numbers and discovered that they arise out of the increasing sequence of odd numbers; for the unity is a square and from it is made the first square, namely 1; to unity is added 3, making the second square, namely 4, with root 2; if to the sum is added the third odd number, namely 5, the third square is created, namely 9, with root 3; and thus sums of square of consecutive odd numbers and a sequence of square always arise together"; this is readily translatable into the following set of equations [3, p4] 1 = 1² 1 + 3 + 5 = 3² 1 + 3 + 5 + 7 = 4² and suggests the general formula, which we would express as: 1 + 3 + ... + (2n-1) = n² These four works and a letter to Theodorus (Epistola ad Magistrum Theodorum [3, xix]) comprise the surviving works of Fibonacci. These however are not the only works composed by Fibonacci.

Other works known to have existed include the Di minor guisa , a book for commercial arithmetic. A discussion of Book X of Euclid's Elements is also known to have existed, in which Fibonacci provided a numerical discussion of irrational numbers. Fibonacci's Mathematics A brief discussion of a few of the mathematical achievements of Leonardo Fibonacci will now be undertaken. Fibonacci showed that there is no pair x and y such that x²+y² and x²-y² are both perfect squares. He also showed that x^4-y^4 can not be a square. Fibonacci defined a special kind of numbers he called a congruum, that obey these rules; a number `k' is a congruum if k = ab(a+b)(a-b) , if a+b is even; or k = 4ab(a+b)(a-b) , if a+b is odd where `a' and `b' are integers. He then shows that x2+h and x2-h can be squares simultaneously only if h is a congruum.

Using this fact he solves the problem set forth by Johannes of Palermo, x2+5 =y2 and x2-5=z2, which rearranges to y2 -x2=5 and x2-z2=5, so y2-x2= x2-z2 , for a=5 and b=4 y2-x2= x2-z2=720, and therefore 492-412=412-312 is the solution, (it is unknown where Fibonacci found the squares of the specific numbers [1, p609]) The discussion of the Fibonacci numbers, while of only minor importance to the works of Fibonacci, has such great notoriety that it would be negligent to ignore it, so a discussion of the Fibonacci numbers and a few interesting applications of the Fibonacci numbers is given in Appendix A. Fibonacci's Main Achievements The most well known of Fibonacci's achievements is definitely the Fibonacci sequence. The Fibonacci sequence was the first recurrence series, other such series as the Lucas series are modelled on it. Fibonacci numbers have applications in modern mathematics.

They are often used in modern computer science, as a part of number theory, and in the counting of mathematical objects. What some consider his most important accomplishment to be the introduction of Hindu-Arabic mathematics into western culture, this is not considered by some to be a great mathematical feat, but rather a compilation of known techniques for a new audience (somewhat equivalent to the statement that Euclid was not so much a mathematician as the author of a successful textbook) In his work, Fibonacci put forth not so much an original exposition (although he showed a certain amount of innovation in some of his solutions) as a compilation of the techniques of Arabic arithmetic and algebra. Interestingly enough Fibonacci was also responsible for the introduction of Arab mathematics to the Arabs as well. Arabic mathematics were originally in use only by scientists and mateticians but not by the Arab businessman [3, xviii].

"[Fibonacci] can rather be credited with the introduction of scientific calculating techniques into general business practice [3, xviii]." These are really only a few of Fibonacci's many achievements, A student both of the algebraic heritage of the east and a scholar of the ancient Greeks, "He joined the theoretical traditions of the Hellenes and the algebraic traditions of the Arabs and established them in Europe. [3,xx]." Fibonacci has been called "the first great mathematician of the Christian West" [1, p611]. It seems an accurate title for the man who had a major part in recognizing the usefulness of and introducing the system the system of numbers we use still today, for the man who has provided a foundation for modern number theory, and many other useful parts of mathematics. Appendix A The Fibonacci Numbers The best known work of Fibonacci concerns the Fibonacci numbers, first examined in the Liber abbaci through the rabbit problem. 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.

From this statement the famous Fibonacci numbers can be derived. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ... where the first two numbers of the series are 1 and 1 and each numbers afterwards is defined as the sum of the two previous terms, Fn=Fn-2+Fn-1. (Though in Fibonacci's sequence the first number was 1 and the second number was two, the first one was assumed) The genealogical tree is shown below: The definition of the Fibonacci numbers is an example of what we refer to today as a recurrence relation. The Fibonacci numbers involve the golden ratio (the solution to the equation x²-x-1=0), which was known already to the ancient Greeks. The ratio of successive Fibonacci numbers appears, on inspection, to converge to a number around 1.618.

If we assume that these ratios do converge to a limit, then for large indices we have Accordingly the limit (assuming it exists) of these ratios, which we will refer to as phi, satisfies the equation phi=1/phi+1, and so phi²-phi-1=0, whose positive solution is the golden ratio. Using this quantity phi, the Fibonacci numbers can be defined explicitly by a formula which is known as Binet's equation: The Fibonacci numbers can be used in a method of calculating pi far more efficient than the method of inscribed and circumscribed polygons. This begins with the Taylor expansion of the arctangent: ... Listen: Mathematician Keith Devlin on "Leonardo of Pisa and the First Personal Computing Revolution" :: News & Events :: Swarthmore College Listen: Mathematician Keith Devlin on "Leonardo of Pisa and the First Personal Computing Revolution" May 10th, 2016 Keith Devlin presents this year's Kaori Kitao Lecture.

Audio: [1 hour 4 sec] During the spring semester, Mathematician Keith Devlin delivered the annual Kaori Kitao lecture on "Leonardo of Pisa and the First Personal Computing Revolution." Based on Devlin’s book The Man of Numbers: Fibonacci’s Arithmetical Revolution (Walker & Co, July 2011) and his co-published companion e-book Leonardo and Steve: The Young Genius Who Beat Apple to Market by 800 Years (Amazon Kindle), the lecture makes the case for the first personal computing revolution taking place not in Silicon Valley in the 1980's but in Pisa in the Thirteenth Century. According to Devlin, the medieval counterpart to Steve Jobs was a young Italian called Leonardo, better known today by the nickname Fibonacci. Thanks to a recently discovered manuscript in a library in Florence, the story of how the little known genius behind the famous name Fibonacci came to launch the modern commercial world can now be told. Devlin is a co-founder and Executive Director of Stanford University's H-STAR institute, a co-founder of the Stanford Media X research network, and a Senior Researcher at CSLI .

He is a World Economic Forum Fellow, a Fellow of the American Association for the Advancement of Science, and a Fellow of the American Mathematical Society. His current research is focused on the use of different media to teach and communicate mathematics to diverse audiences. He also appears as "the Math Guy" on National Public Radio.

Question: By what name was the mathematician 'Leonardo of Pisa' better known?
Answer: Fibonacci