I heard the claim from one European reader that The Arab world invented the zero, and its been downhill ever since. This is false, but unfortunately not an uncommon mistake.
Our numeral system dates back to India during the post-Roman era, but it came to Europe via the medieval Middle East which is why these numbers are called Arabic numbers in many European languages. Yet even Muslims admit that they imported these numerals from India. Calling them Arabic numerals is this therefore deeply misleading. Hindu-Arabic number system could be accepted, but the preferred term should be Indian numerals.
The Maya in Mesoamerica developed a place-value number system with a zero before the Indians, but this great innovation sadly did not influence peoples elsewhere. According to Michael P. Closs in Mathematics Across Cultures: The History of Non-Western Mathematics, There is reason to credit the Maya with the first invention of a zero symbol. It is absent in the surviving epi-Olmec texts but is very common in the Maya inscriptions. Zeros are found in many chronological counts in the Dresden Codex where they occur in positional contexts just as other numerals. Most Maya glyphs come in several variants and the same is true of the zero sign. The zeros in the codices are identifiable as shells and are always painted red. In most cases, the zero shells are stylized and simplified. In the inscriptions, the most common form of the zero is shaped somewhat like a three quarter portion of a Maltese cross.
The Aztecs who were politically dominant in Central Mexico from the 1300s used hand, heart and arrow symbols to represent fractional distances when calculating areas of land. Mesoamerican and especially Mayan mathematics is the one pre-Columbian scientific achievement which compares most favorably to developments in the Old World, but the mainstream development of mathematics happened in the major Eurasian civilizations and the Maya seem to have concentrated their efforts largely in the field of planetary astronomy.
The zero can be used as an empty place indicator, to show that 2106 is different from 216. The ancient Babylonians had a place-value number system with this feature, but base 60. The second use of zero is as a number itself in the form we use it. Some historians of mathematics believe that the Indian use of zero evolved from innovations by Greek astronomers. Symbols for the first nine numbers of our number system have their origins in the Brahmi system of writing in India, which dates back to at least the mid-third century BC. More important than the form of the symbols is the notion of place value, and here the evidence is weaker.
The Chinese had a multiplicative system with the base 10, probably derived from the Chinese counting board. By the fourth century BC the counting board, a checker board with rows and columns, had come into use there. Numbers were represented by little rods made from bamboo or ivory. The abacus was introduced in China around the fourteenth century AD. Somewhere around or before the year 600 AD (the exact place and date remains uncertain) Indians dropped symbols for numbers higher than 9 and began to use symbols for 1 through 9 in our familiar place-value arrangement. Authors James E. McClellan and Harold Dorn speculate whether The appearance of zero within the context of Indian mathematics may possibly be due to specifically Indian religio-philosophical notions of nothingness. This is controversial but worth considering. Ideas have practical consequences, and it sounds plausible that the concept of nothingness would have greater cultural resonance in a country influenced by Hinduism and Buddhism than in Christian-dominated Europe, for example.
The question nevertheless remains why Indians dropped their own multiplicative system and introduced the place-value system, including a symbol for zero. We currently dont know for sure. Victor J. Katz elaborates in his fine A History of Mathematic, Second Edition:
It has been suggested, however, that the true origins of the system in India may be found in the Chinese counting board. Counting boards were portable. Certainly, Chinese traders who visited India brought them along. In fact, since southeast Asia is the border between Hindu culture and Chinese influence, it may well have been the area in which the interchange took place. Perhaps what happened was that the Indians were impressed with the idea of using only nine symbols, but they took for their symbols the ones they had already been using. They then improved the Chinese system of counting rods by using exactly the same symbols for each place value rather than alternating two types of symbols in the various places. And because they needed to be able to write numbers in some form, rather than just have them on the counting board, they were forced to use a symbol, the dot and later the circle, to represent the blank column of the counting board. If this theory is correct, it is somewhat ironic that Indian scientists then returned the favor and brought this new system back to China early in the eighth century.
A decimal place-value system for integers definitely existed in India by the eighth century AD, possibly earlier. Although decimal fractions were used in China, in India there is no early evidence of their use. It was the Muslims who completed the Indian written decimal place-value system by introducing these decimal fractions.
There is evidence of the transmission of pre-Ptolemaic Greek astronomical knowledge to India, possibly along the Roman trade routes. The earliest known Indian work containing trigonometry dates from the fifth century AD. The Gupta period from the fourth to seventh centuries was a golden age for Indian civilization, with a flourishing of art and literature. Astronomers produced a series of textbooks (siddhanta or solutions) covering the basics of astronomy and planetary movements using Greek planetary theory. The Aryabhatiya of the prominent Indian mathematical astronomer Aryabhata (476-550) from 499 was an important work which summarized Hindu mathematics up to that point in time, covering arithmetic, algebra, plane trigonometry and spherical trigonometry. Next to Aryabhata, Brahmagupta (598-ca. 665) was the most accomplished Indian astronomer and mathematician of this age, making advances in algorithms for square roots and the solution of quadratic equations.
As Victor J. Katz writes, in 773 an Indian scholar visited the court of al-Mansur in Baghdad, bringing with him a copy of an Indian astronomical text, quite possibly Brahmaguptas Brahmasphutasiddhanta. The caliph ordered this work translated into Arabic .The earliest available arithmetic text that deals with Hindu numbers is the Kitab al-jamwal tafriq bi hisab al-Hind (Book on Addition and Subtraction after the Method of the Indians) by Muhammad ibn-Musa al-Khwarizmi (ca. 780-850), an early member of the House of Wisdom. Unfortunately, there is no extant Arabic manuscript of this work, only several different Latin versions made in Europe in the twelfth century. In his text al-Khwarizmi introduced nine characters to designate the first nine numbers and, as the Latin version tells us, a circle to designate zero. He demonstrated how to write any number using these characters in our familiar place-value notation. He then described the algorithms of addition, subtraction, multiplication, division, halving, doubling, and determining square roots, and gave examples of their use.
Some Sanskrit works were introduced to Europe via Arabic translations. One Latin manuscript begins with the words Dixit Algorismi, or al-Khwarizmi says. The word algorismi, through some misunderstandings became a term referring to various arithmetic operations and the source of the word algorithm. Zero derives from sifr, Latinized into zephirum. The word sifr itself was an Arabic translation of Sanskrit sunya, meaning empty. The English word sine comes from a series of mistranslations of the Sanskrit jya-ardha (chord-half). Aryabhata frequently abbreviated this term to jya or jiva. When some Hindu works were translated into Arabic, this word was transcribed phonetically into jiba. But since Arabic is a consonantal alphabet usually written without added short vowels, later writers interpreted the consonants jb as jaib, which means bosom or breast. When an Arabic work of trigonometry was translated into Latin, the translator used the equivalent Latin word sinus, which also meant bosom. This Latin word has become our modern English sine.
Rabbi Abraham ben Meir ibn Ezra, or Abenezra (ca. 1090-1167), a Spanish-Jewish philosopher, poet and Biblical commentator, left Spain before 1140 to escape persecution of the Jews by the regime of the Muslim Almohads. He wrote three treatises which helped to bring the Indian symbols to the attention of some of the learned people in Europe, but it took several centuries for Indian numerals to become fully adopted in Europe.
Leonardo of Pisa (ca. 1170-1240), often known as Fibonacci (son of Bonaccio), was an Italian and the first great Western mathematician after the decline of ancient Greek science. The son of a merchant from the city of Pisa with contacts in North Africa, Leonardo himself travelled much in the region. He is most famous for his masterpiece the Liber abbaci or Book of Calculation. The word abbaci (from abacus) does not refer to a computing device but to calculation in general. The first edition appeared in 1202, and a revised one was published in 1228. This work enjoyed a wide European readership and contained rules for computing with the new Indian numerals. The examples were often inspired by examples from Arabic-language treatises, but filtered through Leonardos creative and original genius. Indian numerals faced powerful opposition for generations but were gradually adopted during the Renaissance period, especially by Italian merchants. Their practical advantages compared to the more cumbersome Roman numerals were simply too great to ignore, although Roman numerals are still used for certain limited purposes in the West in the twenty-first century.