A brief biographical sketch of Fibonacci, his life, times and mathematical
achievements.

Contents of this Page

Who
was Fibonacci?

His
names

Fibonacci's
Mathematical Contributions

Introducing
the Decimal Number system into Europe

Roman
Numerals

Arithmetic
with Roman Numerals

The
Decimal Positional System

"Algorithm"

The
Fibonacci Numbers

Did
Fibonacci invent this Series?

Naming
the Series

Fibonacci
memorials to see in Pisa

Fibonacci's
mathematical books

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), 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 North African town of Bugia now called Bougie where wax candles
were exported to France. They are still called "bougies" in French, but the town
is a ruin today says D E Smith (see below).

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 = 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).

SO 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

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).

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!!

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 abovewritten 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?
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 Fn
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 Fn+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 statute 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 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

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.

The
Golden String

Fibonacci Home Page

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Applications of the
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©
1996-2007 Dr Ron
Knott
created 11 March 1998, Latest update: 4 September 2007