held a diplomatic post. As he grew older, Fibonacci
also traveled to Greece, Egypt, Syria, France, and Sicily,
and took special note of the arithmetic systems used by
local merchants of those areas. He became convinced
that the number system used by the Arabs with its
roots from India—the actual way they wrote numbers
and the way they manipulated them to perform calcula-
tions—was far superior to any other arithmetic system
he had encountered, including the clumsy system of
R
OMAN NUMERALS
in use at the time in Europe. Upon
his return to Pisa around the year 1200, Fibonacci
began writing his famous piece. Its aim was to simply
explain the Hindu-Arabic numerals, the role of a
PLACE
-
VALUE SYSTEM
, and illustrate its superior
approach. The text begins simply:
These are the nine figures of the Indians: 9, 8,
7, 6, 5, 4, 3, 2, 1. With these nine figures, and
with this sign 0, … any number may be writ-
ten, as will be shown.
Divided into four sections, the work outlines the meth-
ods of addition, multiplication, subtraction, and divi-
sion. It also discusses fractions (including a discussion
on E
GYPTIAN FRACTIONS
), as well as some geometry
and algebra. (Some parts of the text are written from
right to left, indicating, perhaps, the extent to which
Fibonacci was influenced by Arabic scholars.)
Although complete acceptance of the Hindu-Arabic
system in the West did not occur until about 300 years
later, Fibonacci’s work in this area is recognized as the
first significant step in this direction.
Fibonacci also wrote extensively in the fields of
number theory, trigonometry, and geometry. It is said
that the advances Fibonacci presented in his 1225
piece Liber quadratorum (The book of square num-
bers) were of such interest and value that they sparked
renewed interest in theoretical mathematics and
revived Western mathematics from its slumber during
the Middle Ages.
During his life Fibonacci was recognized as a great
scholar. Word of his abilities reached the Emperor
Frederick II, seated in Palermo, who invited him to
compete against other mathematicians of the day in a
mathematical tournament. Fibonacci correctly solved
all three challenges put before him, garnering him fur-
ther attention and fame. In 1240 Fibonacci was
awarded a salary from the city of Pisa in recognition of
his services to the community.
All of Fibonacci’s texts, and their reproductions,
were written by hand. Copies of Liber abaci still sur-
vive today.
Fibonacci’s name is derived from the shortening of
the Latin filius Bonacci, meaning the son of Bonaccio,
his father’s family name. During his life, Fibonacci was
also known as Leonardo of Pisa or, in Latin, Leonardo
Pisano. Sometimes, Fibonacci also identified himself as
Leonardo Bigollo, following the Tuscan word bigollo
for “a traveler.”
By introducing the Hindu-Arabic numeral to
Europe, his influence on Western mathematics was pro-
found. He died in the city of Pisa, Italy, likely in the
year 1250. (The exact date of death is not known.)
See also F
IBONACCI NUMBERS
.
Fibonacci numbers Any one of the numbers that
appears in the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,
89, …, where each number, after the second, is the sum
of the two preceding numbers, is called a Fibonacci
number. If Fndenotes the nth Fibonacci number, then
we have
Fn= Fn–1 + Fn–2
with F1= F2= 1.
These numbers arise from a famous rabbit-breed-
ing problem described in F
IBONACCI
’s text Liber abaci:
How many rabbits would be produced in the
nth month if, starting from a single pair, any
pair of rabbits of one month produces one pair
of rabbits for each month after the next?
(The initial pair of rabbits, for example, do not pro-
duce another pair of rabbits until month 3. This same
pair produces a new pair for each month thereafter.)
In any month, the totality of rabbits present con-
sists of all pairs of the previous month together with all
the new offspring. The number of offspring equals the
population size of two months previous. Thus the solu-
tion to the problem is the sequence described above.
Any problem whose nth case solution is the sum of
the two previous case solutions produces the Fibonacci
192 Fibonacci numbers