Our number system has been inappropriately called the “Arabic number system”. The correct name is the “Hindu number system”, since it actually originated in India.
Positional number systems
In a positional number system like we use today, a single symbol (like “5”) can be used to mean five, or fifty, or five thousand, depending on its position. It is easy for us to take this number system for granted, unless we compare it to the Roman system of numbers.
Our positional system of arithmetic was completely unheard of in thirteenth century Europe. The Roman system, where V meant “five” and L meant “fifty”, was use for notation by European merchants. However, since even adding two numbers together was very cumbersome with Roman numerals, they used the abacus when they wanted to do arithmetic.
Because of this positional notation, there are relatively simple longhand methods for addition, subtraction, multiplication, and division.
The Babylonians had probably the first positional number system around the 19th century BC. Their number system was, however, a sexagesimal system, being based on 60. Remnants of the Babylonian system remain today in the fact that there are 60 seconds in a minute and 60 minutes in an hour.
The Mayan culture independently developed a positional number system that was based on 20 and 18. This system was in use perhaps as early as 400 BC.
A positional number system was also developed in India as early as 594 AD, where it was used to record a date. Numerous other dates were written in positional notation in the 700’s and 800’s. Historians have disputed all these documents. It is possible that they were forgeries actually written at a later date. The first appearance of Indian positional notation that historians have accepted was in 876 AD.
Of the three positional number systems, the Indian system has survived largely because of a good PR man. The Arab al-Khwarizmi wrote a book entitled “Concerning the Hindu Art of Reckoning” sometime in the early 800’s. This book largely was derived upon the earlier work of Brahmagupta.
The account of the Hindu numbers and their use in calculation by al-Khwarizmi was so clear that he was mistakenly credited for developing the system. As a result, we even see historians of mathematics refer to our number system as the Arabic system. [Rouse Ball, p. 166 and p. 167]
Al-Kwarizmi was a Persian from around 800 AD. He belonged to a school in Baghdad that was responsible for the preservation of much of the earlier science. Greece was the center of science up until about 300 AD. India took over, and then the torch was passed to the Persians, thanks to a very forward looking ruler whose name escapes me right now. (This is all from memory, sorry.) We have this king to thank for the knowledge of Euclid's Elements, for the works of Ptolemy and Aristotle, and for the math contributions of the Hindus.
This Persian school was also the birthplace of Algebra. Al-Kwarizmi wrote the first book on algebra. A word or two from the title became our word "algebra".
Fibonacci wrote the book Liber Abaci in 1202, in which he introduced what he called the Arab system of numbers into Italy. Adelard of Bath and John of Seville also introduced the Hindu system to Europe [Boyer, p. 252]
Enter Leonardo di Pisa, also known as Fibonacci. Today, he is known for inventing the Fibonacci series which has to do with counting generations of rabbits. He had a much more significant contribution to European math and commerce, though. He was studying Arabic texts, and came upon a book by Al-Kwarizmi entitled "On the calculation with Hindu numerals".
Getting back to the story, Leonardo was quite taken with the idea of doing calculating with this notation. He understood that, with the Hindu notation, businessmen would no longer need an abacus to multiply two numbers together. They could do long multiplication like we do today. He translated this book by Al-kwarizmi into Latin to make it available to the businessmen of the day.
Leonardo was careful to credit the book to Al-Kwarizmi. He even included the name of the original author in his Latin title: Algoritmi de numero Indorum. Unfortunately for the history of math, the populace did not understand that "Algorithmi" referred to a person. It was assumed that this word was the name for the method of calculation that was introduced to Europe in this book. As an aside, this misconception is the origin of the word algorithm.
Somewhere along the line, Europe also forgot that the original number system came from India, and not Persia. Today, we incorrectly refer to our numbers as Arabic. It is somewhat more correct to refer to our number system as Hindu-Arabic, as they are sometimes called. The accurate description, though, is "Hindu".
The effect of the Hindu number system on European mathematics cannot be over. In the words of Laplace:
The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.