The Bakhshali Manuscript (Bodleian Library Oxford) describes solutions to problems in arithmetic, algebra and geometry

3rd or 4th century AD manuscript on birch bark found in 1881 near the village of Bakhshali in Peshawar now in Pakistan. Most has been destroyed and only a few scraps of ca 70 leaves have survived.

A formula (in modern terminology) for a square root

A formula (in modern terminology) for a square root

√Q = √(A2 + b) = A + b/2A - (b/2A)2/[2(A + b/2A)]

A formula (in modern terminology) for a square root

√Q = √(A2 + b) = A + b/2A - (b/2A)2/[2(A + b/2A)]

If Q = 41 this gives

A formula (in modern terminology) for a square root

√Q = √(A2 + b) = A + b/2A - (b/2A)2/[2(A + b/2A)]

If Q = 41 this gives

√Q = 6.403138528

A formula (in modern terminology) for a square root

√Q = √(A2 + b) = A + b/2A - (b/2A)2/[2(A + b/2A)]

If Q = 41 this gives

√Q = 6.403138528 6.403124237 correct

result

An early mathematical manuscript, written on birch bark and found in the summer of 1881 near the village of Bakhshali in the Yusufzai subdivision of the Peshawar district (now in Pakistan). A large part of the manuscript had been destroyed and only about 70 leaves of birch bark, of which a few were mere scraps, survived to the time of its discovery. Although its date is uncertain, it is most commonly put at about the third or fourth century AD. and appears to be a commentary on an earlier mathematical work. Among the rules and techniques it sets out for solving problems, mostly in arithmetic and algebra, but also to a lesser extent in geometry and menstruation, is this formula (stated here in modern terms) for calculating the square root of a non-square number Q:

√Q = √(A2 + b) = A + b/2A - (b/2A)2/[2(A + b/2A)]

If Q = 41 (so that A = 6 and b = 5) this gives √Q = 6.403138528, which compares very favorably with the correct result of 6.403124237.

Another interesting piece of mathematics in the manuscript concerns calculating square roots. The following formula is used

√Q = √(A2 + b) = A + b/2A - (b/2A)2/(2(A + b/2A))

This is stated in the manuscript as follows:- In the case of a non-square number, subtract the nearest square number, divide the remainder by twice this nearest square; half the square of this is divided by the sum of the approximate root and the fraction. this is subtracted and will give the corrected root.

Taking Q = 41, then A = 6, b = 5 and we obtain 6.403138528 as the approximation to √41 = 6.403124237. Hence we see that the Bakhshali formula gives the result correct to four decimal places.

The Bakhshali manuscript also uses the formula to compute √105 giving 10.24695122 as the approximation to √105 = 10.24695077.

This time the Bakhshali formula gives the result correct to five decimal places. The following examples also occur in the Bakhshali manuscript where the author applies the formula to obtain approximate square roots: √487

Bakhshali formula gives 22.068076490965Correct answer is 22.068076490713Here 9 decimal places are correct √889

Bakhshali formula gives 582.2447938796899Correct answer is 582.2447938796876Here 11 decimal places are correct

It is interesting to note that Channabasappa [6] derives from the Bakhshali square root formula an iterative scheme for approximating square roots. He finds in [7] that it is 38% faster than Newton's method in giving √41 to ten places of decimals.

Bakhshali formula gives 29.816105242176Correct answer is 29.8161030317511Here 5 decimal places are correct [Note. If we took 889 = 302 - 11 instead of 292 + 48 we would get Bakhshali formula gives 29.816103037078Correct answer is 29.8161030317511Here 8 decimal places are correct] √339009

Bakhshali formula gives 582.2447938796899Correct answer is 582.2447938796876Here 11 decimal places are correct It is interesting to note that Channabasappa [6] derives from the Bakhshali square root formula an iterative scheme for approximating square roots. He finds in [7] that it is 38% faster than Newton's method in giving √41 to ten places of decimals.

OriginsMain article: History of the Arabic numeral systemThe digits 1 to 9 in the Arabic numeral system evolved from the Brahmi numerals, but Brahmi numerals lacked a symbol for 0.By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. By 300 BC, a punctuation symbol (two slanted wedges) was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish (dating from about 700 BC), the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges.[8]

The Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number. Thus numbers like 2 and 120 (2×60), 3 and 180 (3×60), 4 and 240 (4×60), looked the same because the larger numbers lacked a final sexagesimal placeholder. Only context could differentiate them.The first universally accepted inscription containing the use of the 0 glyph is first recorded in the 9th century, in an inscription at Gwalior in Central India dated to 870. By this time, the use of the glyph had already reached Persia, and was mentioned in Al-Khwarizmi's descriptions of Arabic numerals. Documents on copper plates exist, with the same symbol for zero in them, dated back as far as the 6th century CE.[9]

http://en.wikipedia.org/wiki/Arabic_numerals

                                                   

Brahmi numerals (lower row) in India in the 1st century CE

       

Modern-day Arab telephone keypad with two forms of Arabic numerals: Western Arabic/European numerals on the left and Indian numerals on the rightThe numeral system came to be known to both the Persian mathematician Al-Khwarizmi, whose book On the Calculation with Hindu Numerals written about 825 in Arabic, and the Arab mathematician Al-Kindi, who wrote four volumes, "On the Use of the Indian Numerals" (Ketab fi Isti'mal al-'Adad al-Hindi) about 830. Their work was principally responsible for the diffusion of the Indian system of numeration in the Middle East and the West.[10] In the 10th century, Middle-Eastern mathematicians extended the decimal numeral system to include fractions, as recorded in a treatise by Syrian mathematician Abu'l-Hasan al-Uqlidisi in

Arabic numerals [1][2][3] [4] are the ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). They are descended from the pre-Islamic Arabic numeral system developed by Arabian mathematicians,[5] in which a sequence of digits such as "975" is read as a numeral. They were transmitted to Europe in the Middle Ages. The use of Arabic numerals spread around the world through European trade, books and colonialism. Today they are the most common symbolic representation of numbers in the world.The reason the digits are more commonly known as "Arabic numerals" in Europe and the Americas is that they were introduced to Europe in the 10th century by Arabs in North Africa & Middle East, who were then using the digits.This is not to be confused with what the Arabs call the "Hindi numerals", namely the Eastern numerals (٩ - ٨ - ٧ - ٦ - ٥ - ٤ - ٣ - ٢ - ١ - ٠) used in the Middle East, or any of the numerals currently used in Indian languages (e.g. Devanagari: ०.१.२.३.४.५.६.७.८.९).[6]

In English, the term Arabic numerals can be ambiguous. It most commonly refers to the numeral system widely used in Europe and the Americas. Arabic numerals is the conventional name for the entire family of related systems of Arabic. It may also be intended to mean the numerals used by Arabs in North Africa & Middle East, in which case it generally refers to the Eastern Arabic-Indian numerals.The decimal Arabic numeral system was invented around 500 CE. The system was revolutionary by including a zero and positional notation. It is considered an important milestone in the development of mathematics. One may distinguish between this positional system, which is identical throughout the family, and the precise glyphs used to write the numerals, which vary regionally. The glyphs most commonly used in conjunction with the Latin script since early modern times are 0 1 2 3 4 5 6 7 8 9.Although the phrase "Arabic numeral" is frequently capitalized, it is sometimes written in lower case: for instance, in its entry in the Oxford English dictionary.[7] This helps distinguish it from "Arabic numerals" as the East Arabic numerals specific to the Arabs.