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Who was Fibonacci?

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

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).

(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 (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

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

Fibonacci's Mathematical Contributions

Introducing the Decimal Number system into Europe

1 2 3 4 5 6 7 8 9 0

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 -------

Roman Numerals

The Numerals are letters

I = 1, V = 5, X = 10, L = 50, C = 100, D = 500 and M = 1000

The Additive rule

XIII

IIIX

IIXI

Sing a song of sixpence

A pocket full of rye

Four and twenty blackbirds

Baked in a pie...

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

XI

IX

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

The Decimal Positional System

MMIII

IIIMM

IX

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 knitting pattern;

travel instructions;

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!!

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

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.

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

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?

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 Gosp a la (before 1135) and Hemachandra (c. 1150) mentioned the numbers 1, 2, 3, 5, 8, 13, 21, ... explicitly.

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

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

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. The Book of Squares his largest book: an annotated translation into English of Leonardo Fibonaci's 1225 AD version of Liber quadratorum by L E Sigler, 1987, Academic Press, 124 pages.

Starting with a brief biography of Fibonacci, this is an interesting and ingenious book on all sorts of questions about expressing a number as the sum of two, three of four square numbers (or squared fractions).

If we can express a square number also as the sum of two other square numbers then Pythagoras' Theorem tells us that we have three sides of a right-angled triangle and this is Fibonacci's first Proposition. It seems that he was familiar with Euclid's Elements which also contains (Book 10, Proposition 29), Lemma 1) the same method of constructing all sets of three numbers that are the sides of a right-angled triangle. even though Fibonacci does not use the algebraic notation we do today, it is marvellously clear in its desriptions of the processes and algorithms and Sigler's notes show the algebraic notation to explain Fibonacci's process as we would write them today. Another article about this book:

Leonardo of Pisa and his Liber Quadratorum by R B McClenon in American Mathematical Monthly vol 26, pages 1-8. A letter to Master Theodorus, around 1225. Theodorus was a philosopher at the court of the Holy Roman Emporer Frederick II.

There is a very readable outline of the problems in the letter to Master Theodorus in:

Fibonacci's Mathematical Letter to Master Theodorus A F Horodam, Fibonacci Quarterly 1991, vol 29, pages 103-107. Practica geometriae, 1220. A book on geometry. Flos, 1225

The most comprehensive translation of the manuscripts of the 5 works above is:

Scritti di Leonardo Pisano B Boncompagni, 2 volumes, published in Rome in 1857 (vol 1) and 1862 (vol 2).



References to Fibonacci's Life and Times

© 1996-2009 Dr Ron Knott

created 11 March 1998, Latest update: 28 September 2009