# Italy

### Mathematicians

Leonardo Fibonacci was a famous Italian mathematician who lived from 1170-1240 A.D. When he was a boy, he traveled widely on business in the Middle East with his father. While traveling he learned a lot about Greek and Arabic mathematics. Fibonacci was also introduced to many different mathematical ideas used throughout the world. He put these ideas into books so people could learn them without having to travel as he had.

This mathematician also discovered a sequence of numbers called the Fibonacci Sequence or Fibonacci Numbers. This sequence contains the numbers 1,1,2,3,5,8,13,..... Each number after the first two numbers equals the sum of the two numbers before it. Here's an example: 1+1=2, 1+2=3, 2+3=5, 5+3=8, 8+5=13... Leonardo Fibonacci also was the first person to discover the Golden Rectangle, Golden Ratio and the Golden Spiral. Leonardo Fibonacci's sequence, rectangle ratio and spiral have been used and studied people by people all over the world.

In his book, Liber Abaci , Fibonacci made a detailed account of the mathematical experiences on his Mediterranean travels. Fibonacci begins the book with a description of the mathematics inherent in India:

The nine Indian figures are:

9 8 7 6 5 4 3 2 1

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: (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?

### Problem 3:

Solve in rational numbers the pair of equations ,  .

### Interesting Fact

Fibonacci is responsible for our using a bar to separate the numerator and denominator of a fraction. Otherwise, would simply be written vertically as a 2 “over” 3 without the bar! In addition, Fibonacci retained the Arabic right-to-left convention of reading mixed fractions: instead of . 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.

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.

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 4

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?

### Interesting Link (Fibonacci Numbers and Nature):

Try solving the International Color Challenge!

Reference:

A.F. Horadam, “Eight hundred years young,” The Australian Mathematics Teacher 31 (1975) 123-134.