Definitions

# History of the Hindu-Arabic numeral system

The Hindu-Arabic numeral system is a place-value numeral system: the value of a digit depends on the place where it appears; the '2' in 205 is ten times greater than the '2' in 25. It requires a zero to handle the empty powers of ten (as in "205").

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.

## Decimal System

An early decimal system was clearly in use by the inhabitants of the Indus valley civilization by 3000 BC. Excavations at both Harappa and Mohenjo Daro reveal decimal weights belonging to "two series both being decimal in nature with each decimal number multiplied and divided by two, giving for the main series ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500." Also, marked rulers at Lodhar (?Lothal) reveal gradations of 1.32 inches (3.35 centimetres), ten of which are 13.2 inches, possibly something akin to a "foot" (similar measures exist in other parts of Asia and beyond). Markings on these and other texts reveal a number system with symbols for the numbers one through nine, and separate symbols for 10, 20, 100; thus the decimal system is highly developed though place-value is not 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.

## Positional notation

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.

## Rational numbers

In Sanskrit literature, fractions, or rational numbers were always expressed by an integer followed by a fraction. When the integer is written on a line, the fraction is placed below it and is itself written on two lines, the numerator called amsa part on the first line, the denominator called cheda “divisor” on the second below. If the fraction is written without any particular additional sign, one understands that it is added to the integer above it. If it is marked by a small circle or a cross (the shape of the “plus” sign in the West) placed on its right, one understands that it is subtracted from the integer. For example, Bhaskara I writes

`६   १   २`
`१   १   १०`
`४  ५   ९`

That is,

`6   1   2`
`1   1   1०`
`4   5   9`

to denote 6+1/4, 1+1/5, and 2–1/9

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:

"I will omit all discussion of the science of the Indians, ... , of their subtle discoveries in astronomy, discoveries that are more ingenious than those of the Greeks and the Babylonians, and of their valuable methods of calculation which surpass description. I wish only to say that this computation is done by means of nine signs. If those who believe, because they speak Greek, that they have arrived at the limits of science, would read the Indian texts, they would be convinced, even if a little late in the day, that there are others who know something of value."

According to al-Qifti's chronology of the scholars :

"... a person from India presented himself before the Caliph al-Mansur in the year [776 AD] who was well versed in the siddhanta method of calculation related to the movement of the heavenly bodies, and having ways of calculating equations based on the half-chord [essentially the sine] calculated in half-degrees ... This is all contained in a work ... from which he claimed to have taken the half-chord calculated for one minute. Al-Mansur ordered this book to be translated into Arabic, and a work to be written, based on the translation, to give the Arabs a solid base for calculating the movements of the planets ..."

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:

"It seems plausible that it drifted gradually, probably before the seventh century, through two channels, one starting from Sind, undergoing Persian filtration and spreading in what is now known as the Middle East, and the other starting from the coasts of the Indian Ocean and extending to the southern coasts of the Mediterranean."

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 [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.

## Evolution of symbols

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:

• 976

The first Arabic numerals in Europe appeared in the Codex Vigilanus in the year 976.

• 1202

Fibonacci, an Italian mathematician who had studied in Béjaïa (Bougie), Algeria, promoted the Arabic numeral system in Europe with his book Liber Abaci, which was published in 1202.

• 1482

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.)

• 1549

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.

Even in many countries in languages which have their own numeral systems, the European Arabic numerals are widely used in commerce and mathematics.

## Impact on Mathematics

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:

"It is India that gave us the ingenuous method of expressing all numbers by the means of ten symbols, each symbol receiving a value of position, as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit, but its very simplicity, the great ease which it has lent to all computations, puts our arithmetic in the first rank of useful inventions, and we shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest minds produced by antiquity."

Tobias Dantzig, the father of George Dantzig, had this to say in Number:

"This long period of nearly five thousand years saw the rise and fall of many a civilization, each leaving behind it a heritage of literature, art, philosophy, and religion. But what was the net achievement in the field of reckoning, the earliest art practiced by man? An inflexible numeration so crude as to make progress well nigh impossible, and a calculating device so limited in scope that even elementary calculations called for the services of an expert [...] Man used these devices for thousands of years without contributing a single important idea to the system [...] Even when compared with the slow growth of ideas during the dark ages, the history of reckoning presents a peculiar picture of desolate stagnation. When viewed in this light, the achievements of the unknown Hindu, who some time in the first centuries of our era discovered the principle of position, assumes the importance of a world event."