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
Fibonacci
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 Numerals are letters
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 = 1000You 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 sixpenceAbove 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.
A pocket full of rye
Four and twenty blackbirds
Baked in a pie...
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
Arithmetic was not easy in the Roman system:CLXXIIII added to XXVIII is CCII CLXXIIII less XXVIII is CXXXXVI
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).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 beAl-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.
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a method followed by humans, for example:a cooking recipe;
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 YearHe then goes on to solve and explain the solution:
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.
Because the above written pair in the first month bore, you will double it; there will be two pairs in one month. | beginning 1 first 2 second 3 third 5 fourth 8 fifth 13 sixth 21 seventh 34 eighth 55 ninth 89 tenth 144 eleventh 233 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. 
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