A brief biographical sketch of Fibonacci, his life, times and mathematical
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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.]
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").
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.]
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
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
The the Math Forum's
archives of the
History of Mathematics discussion group
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.
The Numerals are letters
The method in use in Europe until then used the Roman
I = 1,
V = 5,
X = 10,
L = 50,
C = 100,
D = 500 and
M = 1000
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
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
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
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
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
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.
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.
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.
Earlier the Persian author
Abu ‘Abd Allah,
Mohammed ibn Musa al-Khwarizmi
(usually abbreviated to
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
He was an astromomer to the caliph at Baghdad (now in Iraq).
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
part of the former USSR, north of Iran, which gained its independence in 1991.
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
The USA Library of Congress
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;
or mechanically by machines (such as placing chips and components at correct
places on a circuit board to go inside your TV)
a knitting pattern;
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!
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 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
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
He 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.
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.
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
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.
- The Book of Squares
- his largest book:
an annotated translation into English of Leonardo Fibonaci's 1225 AD version of
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
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
Leonardo of Pisa and the New Mathematics of the Middle Ages J Gies, F Gies,
Crowell press, 1969.
The Autobiography of Leonardo Pisano R E Grimm, in
Fibonacci Quarterly, vol 11, 1973, pages 99-104 with corrections
on pages 162 and 168.
800 Years young A F Horodam in
Australian Mathematics Teacher vol 31, 1975, pages 123-134.
© 1996-2009 Dr Ron Knott
created 11 March 1998, Latest update: 28 September 2009