**THE FAMOUS FIBONACCI NUMBERS**

Now that we have established the beginnings of the number zero, let's see how the numbers that we use today evolved and spread throughout the world. As mentioned above, they originated in India—invented there by mathematicians between the first and fourth centuries CE—and were brought into the European world in the year 1202 by the famous mathematician Leonardo of Pisa, better known today as Fibonacci. As mentioned above, Fibonacci wrote a book called Liber Abaci. It contained many mathematical problems, and the first words of its introduction were:

The nine Indian figures are: 9 8 7 6 5 4 3 2 1.

With these nine figures, and with the sign 0, which the Arabs call zephyr, any number whatsoever is written, as demonstrated below. A number is a sum of units, and through the addition of them the number increases by steps without end. First one composes those numbers, which are from one to ten. Second, from the tens are made those numbers, which are from ten up to one hundred. Third, from the hundreds are made those numbers, which are from one hundred up to one thousand…. And thus, by an unending sequence of steps, any number whatsoever is constructed by joining the preceding numbers. The first place in the writing of the numbers is at the right. The second follows the first to the left.

Fibonacci encountered these numbers during his travels on the Barbary Coast of Africa, where he worked closely with Arab mathematicians. Today, these numerals are referred to as Hindu-Arabic numerals, depicting their path to Europe. Despite their relative facility, these numerals were not widely accepted by merchants who were suspicious of those who knew how to use them. Such merchants were simply afraid of being cheated. The numerals first began to be used fifty years after the publication of Fibonacci's book, but still not very extensively. We can safely say that it took the same three hundred years for these numerals to catch on as it did for the leaning tower of Pisa to be completed.

Interestingly, Liber Abaci also contains simultaneous linear equations. Many of the problems that Fibonacci considers, however, were similar to those appearing in Arab sources. This does not detract from the value of the book, since it is the collection of solutions to these problems that makes it a major contribution to our development of mathematics. As a matter of fact, a number of mathematical terms—common in today's usage—were first introduced in Liber Abaci. Fibonacci referred to factus ex multiplicatione, and from this first sighting of the phrase, we now speak of the “factors of a number” or the “factors of a multiplication.” Another example of words whose introduction into the current mathematics vocabulary seems to stem from this famous book are “numerator” and “denominator.”

Before we go further with these common numerals, it is worthwhile to see what predated them. This we will see when we look at the ancient Egyptian number system.