In his book, Liber Abbaci, Fibonacci made a detailed account of the mathematical
experiences on his Mediterranean travels. Fibonacci began the book with a description
of the mathematics inherent in India:
The nine Indian figures are:
987654321
With these nine figures, and with the sign 0…any number may be written, as is
demonstrated below.
Thus, Fibonacci discussed the nature of Hindu-Arabic numerals and their applications to
calculations, commercial problems, bartering, and interest and money-changing.
Fibonacci also discussed series and proportions, extraction of square and cube roots, the
Rule of False Position, the Method of Casting out Nines, and other techniques employed
by Hindu and Arab mathematicians.
Problem 1: (The Lion and the Pit)
A pit was 50 handbreadths in depth: A lion climbed up the pit 1/7 handbreadth
every day and fell back 1/9 handbreadth. How long would it take him to get out
of the pit?
Problem 2: (Birds and Towers)
Two birds start flying from the tops of two towers 50 feet apart, one 30 feet high,
the other 40 feet high, starting at the same time and at the same rate. They reach
the center of a fountain between the towers at the same time. How far is the
fountain from each tower?
Problem 3: (An Inheritance)
A man whose end was approaching summoned his sons and said: “Divide my
money as I shall prescribe.” To his eldest son, he said, “You are to have 1 bezant
and a seventh of what is left.” To his second son he said, “Take 2 bezants and a
seventh of what remains.” To the third son, “You are to take 3 bezants and a
seventh of what is left.” Thus he gave each son 1 bezant more than the previous
son and a seventh of what remained, and to the last son all that was left. After
following their father’s instructions with care, the sons found that they had shared
their inheritance equally. How many sons were there, and how large was the
estate?
Problem 4: (A Voyage)
A certain man doing business in Lucca doubled his money there, and then spent
12 denarii. Thereupon, leaving he went to Florence; there he also doubled his
money, and spent 12 denarii. Returning to Pisa, he there doubled his money and
spent 12 denarii, nothing remaining. How much money did he have in the
beginning?
Interesting Fact:
Fibonacci is responsible for our using a bar to separate the numerator and
2
denominator of a fraction. Otherwise, would simply be written vertically as a
3
2 “over” 3 without the bar! In addition, Fibonacci retained the Arabic right-to-left
64 64
convention of reading mixed fractions: 5 instead of 5 .
257 257
Another type of problem that Fibonacci discussed in Liber Abbaci was the Chinese
Remainder Problem. If you visit China, you will learn that it was the Chinese
mathematician, Sun Zi, who posed the earliest known occurrence of the Chinese
Remainder Theorem.
Problem 5:
(i) Find a number which is divisible by 7 and which gives a remainder of 1
when divided by 2, 3, 4, 5, or 6.
(ii) Find a number which is a multiple of 7 having remainders of 1, 2, 3, 4, and
5 when divided by 2, 3, 4, 5 and 6
There are also connections between Fibonacci and Egypt as well as Greece. In his
books, Liber Quadratorum and Flos, Fibonacci solved problems of the type attributed to
the mathematician Diophantus (Diophantine problems). Fibonacci also produced the
mathematical results of Pythagoras.
Problem 6:
Find a square number which, being increased or diminished by 5, gives a square
number.
Extension to Problem 6:
Find a number which, being added to, or subtracted from, a square number, leaves
in either case a square number.
Problem 7:
Find a number of the form 4 xyx y x y which is divisible by 5, the quotient
being a square. (Fibonacci calls a number of this form a congruum).
Problem 8:
Solve in rational numbers the pair of equations x 2 x u 2 , x 2 x v 2 .
Problem 9:
Solve x3 2 x 2 10x 20 .
Problem 10:
Three men owned some money, their shares being ½, 1/3, and 1/6. Each took
some money at random until none was left. The first man then returned ½ of what
he had taken, the second 1/3 and the third 1/6. When the money now in the pile
was divided equally among the men, each possessed what he was entitled to.
How much money was in the original store, and how much did each man take?
Perhaps the contribution to mathematics that bears his name and which Fibonacci is
most famous for is the Fibonacci sequence. In Liber Abbaci, Fibonacci posed the
following problem which requires you to derive this sequence and answer a question
related to it.
Problem 11:
A certain man put a pair of rabbits in a place surrounded by a wall. How many
pairs of rabbits can be produced from that pair in a year if it is supposed that
every month each pair begets a new pair which from the second month on
becomes productive?
Problem 11 Answer: 377
Fibonacci Numbers and Nature:
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html