| Numeral systems by culture | |
|---|---|
| Hindu-Arabic numerals | |
| Western Arabic (Hindu numerals) Eastern Arabic Indian family Tamil |
Burmese Khmer Lao Mongolian Thai |
| East Asian numerals | |
| Chinese Japanese Suzhou |
Korean Vietnamese Counting rods |
| Alphabetic numerals | |
| Abjad Armenian Āryabhaṭa Cyrillic |
Ge'ez Greek Georgian Hebrew |
| Other systems | |
| Aegean Attic Babylonian Brahmi Egyptian Etruscan Inuit |
Kharosthi Mayan Quipu Roman Sumerian Urnfield |
| List of numeral system topics | |
| Positional systems by base | |
| Decimal (10) | |
| 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 20, 24, 30, 36, 60, 64 | |
| Non-positional system | |
| Unary numeral system (Base 1) | |
| List of numeral systems | |
The Hindu-Arabic numeral system is a decimal place-value numeral system. It requires a zero to handle the empty powers of ten (as in "205").[1]
Its glyphs are descended from the Indian Brahmi numerals. The full system emerged by the 8th to 9th centuries, and is first described in 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).[2] Today the name Hindu-Arabic numerals is usually used.
Evidence of early use of a zero glyph may be present in Bakhshali manuscript, a text of uncertain date, possibly a copy of a text composed as early as the 3rd century.
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Historians trace modern numerals in most languages to the Brahmi numerals, which were in use around the middle of the third century BC.[3] 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.[3]
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.[3] Beginning around 7th century, the Gupta numerals evolved into the Nagari numerals.
Comparison of the computation in Kitab al-Fusul fi al-Hisab al Hindi (925) by al-Uqlidisi, and another Latin translation of the Arab manuscript written by the Persian mathematician Khwarizmi (825), uncovered an almost identical algorithm for multiplication and division with the rod calculus described in Mathematical Classic of Sun Zi.[6] In the case of division, the algorithm described by Khwarizmi and algorithm described by Sun Zi four hundred years earlier, are completely identical to the last detail: exactly the same three tier layout, exactly the same assignment of dividend to the middle row, the same assignment of smaller divisor to the bottom row, padded with blank(!) but not "0" to the right, and quotient to top row padded with blanks(!) but not "0"s; identical alignment of the most significant digit, exactly the same way of calculating from left to right, exactly the same way of shifting divisor to the right one position after each step, up to presentation of the remainder in the form of counting rod fraction.[7] Too identical to be explained with independent development. Furthermore, moving material rods on the counting board to the right is a simple matter, while moving written numbers right one step at each stage "is not conducive to a written system", as Lam Lay Yong put it[8].
Tenth century Persian mathematician Kushyar ibn Labban's division algorithm (described in his book Principles of Hindu Reckoning) is also identical to Sunzi division 500 hundred years earlier.[9] The similarity between ibn Labban's square root algorithm and Sunzi's square root algorithm is also "striking".[10]
Chinese Historian of mathematics Lam Lay Yong claims that "The fact that Arabs and the Chinese had identical expression of fractions, identical arithmetic procedures and identical expression of numerals cannot be dismissed as mere coincidence. Given that the Chinese had evolved all these forms and procedures at a significant earlier date, this inevitably points to the Chinese origin of the Hindu Arabic numeral system."[11]
There is indirect evidence that the Indians developed a positional number system as early as the 1st century CE.[3] The Bakhshali manuscript (c. third century 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.[3] 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.[12] Indian books dated to this period are able to denote numbers in the hundred thousands using a place value system.[13] 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,[3] 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.[3] Evidence suggests that a dot had been used in earlier Indian manuscripts to denote an empty place in positional notation. [1]. 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.).[3][14]
The oldest known text to use zero is the Jain text from India entitled the Lokavibhaga , dated 458 AD.[15] Ifrah wrote that a sentence in Lkavibhaga "panchabhyah khalu shunyebhyah param dve sapta chambaram ekam trini cha rupam cha" meant "five voids,then two and seven, the sky, one and three and the form" was the expression of the number 13107200000, was the earliest place value decimal number with the concept of zero.[16]
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.[17]
Before the rise of the Arab Empire, the Hindu-Arabic numeral system was already moving West and was mentioned in Syria in 662 AD by the Nestorian scholar Severus Sebokht who wrote the following:
According to al-Qifti's chronology of the scholars [3]:
The work was most likely to have been Brahmagupta's Brahmasphutasiddhanta (Ifrah) [4] (The Opening of the Universe) which was written in 628 [5]. 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. [6]
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.[18][19] 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 [2]) "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 evolution of the numerals in early Europe is shown below: The French scholar J.E. Montucla created this table “Histoire de la Mathematique”, published in 1757:
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.
Even in many countries in languages which have their own numeral systems, the European Arabic numerals are widely used in commerce and mathematics.
The significance of the development of the positional number system is 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[20][21]: