Last week, I wrote about the Arab mathematician Al-Khwarizmi and the origin of the word algebra. The European and American cultures received much more than just a word from Arab mathematicians. In the 12th century, a man named Guilielmo Bonacci lived in Bejaia, Algeria, where he represented the interests of the traders of Pisa, Italy. His son, Leonardo, was tutored in the Hindu-Arabic mathematics that Al-Khwarizmi had written about several hundred years before. Leonardo, who became known as Fibonacci, wrote in his book Liber Abaci:
When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians’ nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it, for whatever was studied by the art in Egypt, Syria, Greece, Sicily and Provence, in all its various forms.
It was this book, Liber Abaci, written in 1202, that led to the adoption of the Hindu-Arabic numeral system by the Europeans. At the time, Europeans were still using Roman numerals, which are cumbersome for calculation. The Hindu-Arabic numerals and the system of calculation that goes along with them are very useful in both business and science.
The second section of Liber abaci contains a large collection of problems aimed at merchants. They relate to the price of goods, how to calculate profit on transactions, how to convert between the various currencies in use in Mediterranean countries, and problems which had originated in China.
In 1478, the portions of Liber Abaci that dealt with trade and currency conversion were translated into Italian, so that people who didn’t read Latin could understand the ideas as well. This book, known as the Treviso Arithmetic, spread these ideas to a much wider audience than was possible through a book written in Latin. This allowed the Italian merchants to keep much better accounts, enabling trade to flourish.
Previously, calculations had been made using an abacus, but with the Hindu-Arabic numbers and calculation system, people had a much easier time keeping track of their money. Before the advent of the calculation system, merchants had kept a table in their business that was used as a makeshift abacus in order tally up sales totals. This table was known as a counter. Although the counters in today’s businesses are not directly used to tally things up, the machines that do this job are kept on the “counter.”
Fibonacci is probably best known for a series of numbers that bears his name – the “Fibonacci Series.” This series is derived from a problem in Liber Abaci and is created by adding the two numbers in the series to obtain the next. It starts with ones, which are added to get 2. Then the 1+2=3, then 2+3=5, and 3+5=8 and so on. The series begins 1,1,2,3,5,8,13, 21,… There are many applications where this seemingly trivial series of numbers is useful.
One of the more surprising uses of the Fibonacci Series is in stock trading, where investors use the ratio of Fibonacci Numbers to analyze the behavior of stock, bond or commodity markets.