The numeral system was developed in ancient India, and was well established by the time of the Bakhshali manuscript (ca. 3d c. CE). Despite its Indian origins it was initially known in the West as Arabic numerals because of its introduction to Europe through Arabic texts such as Al-Khwarizmi's On the Calculation with Hindu Numerals (ca. 825), and Al-Kindi's four volume work On the Use of the Indian Numerals (ca. 830). Today the name Hindu-Arabic numerals is usually used.
Linguistic comparison among Indo-European languages (ca. 3000 BC), shows a decimal enumeration system . In early Vedic texts, composed between 2500 BC and 1800 BC, we find Sanskrit number words not only for counting numbers in very large ranges, ranging up to 1019, with some puranas referring to numbers as large as 1062.
Historians trace modern numerals in most languages to the Brahmi numerals, which were in use around the middle of the third century BC. The place value system, however, evolved later. The Brahmi numerals have been found in inscriptions in caves and on coins in regions near Pune, Mumbai, and Uttar Pradesh. These numerals (with slight variations) were in use over quite a long time span up to the 4th century AD.
During the Gupta period (early 4th century AD to the late 6th century AD), the Gupta numerals developed from the Brahmi numerals and were spread over large areas by the Gupta empire as they conquered territory . Beginning around 7th century, the Gupta numerals evolved into the Nagari numerals.
There is indirect evidence that the Babylonians had a place value system as early as the 19th century BC, to the base 60, with a separator mark in empty places. This separator mark never was used at the end of a number, and it was not possible to tell the difference between 2 and 20. This innovation was brought about by Brahmagupta of India. Further, the Babylonian place value marker did not stand alone, as per the Indian "0".
There is indirect evidence that the Indians developed a positional number system as early as the first century CE. The Bakhshali manuscript (c. 3d c. BCE) uses a place value system with a dot to denote the zero, which is called shunya-sthAna, "empty-place", and the same symbol is also used in algebraic expressions for the unknown (as in the canonical x in modern algebra). However, the date of the Bakhshali manuscript is hard to establish, and has been the subject of considerable debate. The oldest dated Indian document showing use of the modern place value form is a legal document dated 346 in the Chhedi calendar, which translates to 594 CE. While some historians have claimed that the date on this document was a later forgery, it is not clear what might have motivated it, and it is generally accepted that enumeration using the place-value system was in common use in India by the end of the 6th century. . Indian books dated to this period are able to denote numbers in the hundred thousands using a place value system. Many other inscriptions have been found which are dated and make use of the place-value system for either the date or some other numbers within the text , although some historians claim these to also be forgeries.
In his seminal text of 499, Aryabhata devised a positional number system without a zero digit. He used the word "kha" for the zero position.. Evidence suggests that a dot had been used in earlier Indian manuscripts to denote an empty place in positional notation. The same documents sometimes also used a dot to denote an unknown where we might use x. Later Indian mathematicians had names for zero in positional numbers yet had no symbol for it.
The use of zero in these positional systems are the final step to the system of numerals we are familiar with today. The first inscription showing the use of zero which is dated and is not disputed by any historian is the inscription at Gwalior dated 933 in the Vikrama calendar (876 CE.) .
The oldest known text to use zero is the Jain text from India entitled the Lokavibhaaga , dated 458 AD.
The first indubitable appearance of a symbol for zero appears in 876 in India on a stone tablet in Gwalior. Documents on copper plates, with the same small o in them, dated back as far as the sixth century AD, abound.
६ १ २
१ १ १०
४ ५ ९
6 1 2
1 1 1०
4 5 9
to denote 6+1/4, 1+1/5, and 2–1/9
The work was most likely to have been Brahmagupta's Brahmasphutasiddhanta (Ifrah) (The Opening of the Universe) which was written in 628 Irrespective of whether Ifrah is right, since all Indian texts after Aryabhata's Aryabhatiya used the Indian number system, certainly from this time the Arabs had a translation of a text written in the Indian number system. In his text The Arithmetic of Al-Uqlîdisî (Dordrecht: D. Reidel, 1978), A.S. Saidan's studies were unable to answer in full how the numerals reached the Arab world:
Al-Uqlidisi developed a notation to represent decimal fractions. The numerals came to fame due to their use in the pivotal work of the Persian mathematician Al-Khwarizmi, whose book On the Calculation with Hindu Numerals was written about 825, and the Arab mathematician Al-Kindi, who wrote four volumes (see ) "On the Use of the Indian Numerals" (Ketab fi Isti'mal al-'Adad al-Hindi) about 830. They, amongst other works, contributed to the diffusion of the Indian system of numeration in the Middle-East and the West.
The French scholar J.E. Montucla created this table “Histoire de la Mathematique”, published in 1757:
The first Arabic numerals in Europe appeared in the Codex Vigilanus in the year 976.
The system did not come into wide use in Europe, however, until the invention of printing (See, for example, the 1482 Ptolemaeus map of the world printed by Lienhart Holle in Ulm, and other examples in the Gutenberg Museum in Mainz, Germany.)
These are correct format and sequence of the “modern numbers” in titlepage of the Libro Intitulado Arithmetica Practica by Juan de Yciar, the Basque calligrapher and mathematician, Zaragoza 1549.
In the last few centuries, the European variety of Arabic numbers was spread around the world and gradually became the most commonly used numeral system in the world.
The significance of the development of the positional number system is probably best described by the French mathematician Pierre Simon Laplace (1749 - 1827) who wrote:
Tobias Dantzig, the father of George Dantzig, had this to say in Number: