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67-584: Brahmagupta ( c. 598 – c. 668 CE ) was an Indian mathematician and astronomer . He is the author of two early works on mathematics and astronomy : the Brāhmasphuṭasiddhānta (BSS, "correctly established doctrine of Brahma ", dated 628), a theoretical treatise, and the Khandakhadyaka ("edible bite", dated 665), a more practical text. In 628 CE, Brahmagupta first described gravity as an attractive force, and used
134-412: A / c + b / d × a / c = a ( d + b ) / cd ; and a / c − b / d × a / c = a ( d − b ) / cd . Brahmagupta then goes on to give the sum of the squares and cubes of the first n integers. 12.20. The sum of
201-596: A slippery slope scenario in his style guide that, "if we do end by casting aside the AD/BC convention, almost certainly some will argue that we ought to cast aside as well the conventional numbering system [that is, the method of numbering years] itself, given its Christian basis." Some Christians are offended by the removal of the reference to Jesus, including the Southern Baptist Convention . The abbreviation BCE, just as with BC, always follows
268-441: A distance d from the top of a mountain of height m , and then travels in a straight line to a city at a horizontal distance mx from the base of the mountain, travels the same distance as one who descends vertically down the mountain and then travels along the horizontal to the city. Stated geometrically, this says that if a right-angled triangle has a base of length a = mx and altitude of length b = m + d , then
335-400: A formula useful for generating Pythagorean triples : 12.39. The height of a mountain multiplied by a given multiplier is the distance to a city; it is not erased. When it is divided by the multiplier increased by two it is the leap of one of the two who make the same journey. Or, in other words, if d = mx / x + 2 , then a traveller who "leaps" vertically upwards
402-626: A matter of convenience. There is so much interaction between people of different faiths and cultures – different civilizations, if you like – that some shared way of reckoning time is a necessity. And so the Christian Era has become the Common Era. Adena K. Berkowitz, in her application to argue before the United States Supreme Court , opted to use BCE and CE because, "Given the multicultural society that we live in,
469-419: A multiplier and increased or diminished by an arbitrary [number]. The product of the first [pair], multiplied by the multiplier, with the product of the last [pair], is the last computed. 18.65. The sum of the thunderbolt products is the first. The additive is equal to the product of the additives. The two square-roots, divided by the additive or the subtractive, are the additive rupas . The key to his solution
536-399: A negative is positive; a zero divided by zero is zero; a positive divided by a negative is negative; a negative divided by a positive is [also] negative. 18.35. A negative or a positive divided by zero has that [zero] as its divisor, or zero divided by a negative or a positive [has that negative or positive as its divisor]. The square of a negative or positive is positive; [the square] of zero
603-553: A number in its own right, rather than as simply a placeholder digit in representing another number as was done by the Babylonians or as a symbol for lack of quantity as was done by Ptolemy and the Romans . In chapter eighteen of his Brāhmasphuṭasiddhānta , Brahmagupta describes operations on negative numbers. He first describes addition and subtraction, 18.30. [The sum] of two positives is positives, of two negatives negative; of
670-715: A period of 138 years in which the traditional BC/AD dating notation was used. BCE/CE is used by the College Board in its history tests, and by the Norton Anthology of English Literature . Others have taken a different approach. The US-based History Channel uses BCE/CE notation in articles on non-Christian religious topics such as Jerusalem and Judaism . The 2006 style guide for the Episcopal Diocese Maryland Church News says that BCE and CE should be used. In June 2006, in
737-413: A positive and a negative [the sum] is their difference; if they are equal it is zero. The sum of a negative and zero is negative, [that] of a positive and zero positives, [and that] of two zeros zero. [...] 18.32. A negative minus zero is negative, a positive [minus zero] is positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it
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#1732765949109804-461: A recurrence relation for generating solutions to certain instances of Diophantine equations of the second degree such as Nx + 1 = y (called Pell's equation ) by using the Euclidean algorithm . The Euclidean algorithm was known to him as the "pulverizer" since it breaks numbers down into ever smaller pieces. The nature of squares: 18.64. [Put down] twice the square-root of a given square by
871-480: Is a direct reference to Jesus as Lord . Proponents of the Common Era notation assert that the use of BCE/CE shows sensitivity to those who use the same year numbering system as the one that originated with and is currently used by Christians , but who are not themselves Christian. Former United Nations Secretary-General Kofi Annan has argued: [T]he Christian calendar no longer belongs exclusively to Christians. People of all faiths have taken to using it simply as
938-410: Is astronomy, but it also contains key chapters on mathematics, including algebra, geometry, trigonometry and algorithmics, which are believed to contain new insights due to Brahmagupta himself. Later, Brahmagupta moved to Ujjaini , Avanti , a major centre for astronomy in central India. At the age of 67, he composed his next well-known work Khanda-khādyaka , a practical manual of Indian astronomy in
1005-404: Is his formula for cyclic quadrilaterals . Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure's area, 12.21. The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral. The accurate [area] is the square root from the product of the halves of the sums of
1072-485: Is in particularly common use in Nepal in order to disambiguate dates from the local calendar, Bikram or Vikram Sambat. Disambiguation is needed because the era of the local calendar is quite close to the Common Era. In 2002, an advisory panel for the religious education syllabus for England and Wales recommended introducing BCE/CE dates to schools, and by 2018 some local education authorities were using them. In 2018,
1139-528: Is not known and it is possible that both Greek and Indian syncopation may be derived from a common Babylonian source. The four fundamental operations (addition, subtraction, multiplication, and division) were known to many cultures before Brahmagupta. This current system is based on the Hindu–Arabic numeral system and first appeared in the Brāhmasphuṭasiddhānta . Brahmagupta describes multiplication in
1206-406: Is to be added. He goes on to describe multiplication, 18.33. The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero. But his description of division by zero differs from our modern understanding: 18.34. A positive divided by a positive or a negative divided by
1273-484: Is zero. That of which [the square] is the square is [its] square root. Here Brahmagupta states that 0 / 0 = 0 and as for the question of a / 0 where a ≠ 0 he did not commit himself. His rules for arithmetic on negative numbers and zero are quite close to the modern understanding, except that in modern mathematics division by zero is left undefined . In chapter twelve of his Brāhmasphuṭasiddhānta , Brahmagupta provides
1340-605: The karana category meant to be used by students. Brahmagupta died in 668 CE, and he is presumed to have died in Ujjain. Brahmagupta composed the following treatises: Brahmagupta's mathematical advances were carried on further by Bhāskara II , a lineal descendant in Ujjain, who described Brahmagupta as the ganaka-chakra-chudamani (the gem of the circle of mathematicians). Prithudaka Svamin wrote commentaries on both of his works, rendering difficult verses into simpler language and adding illustrations. Lalla and Bhattotpala in
1407-573: The Chavda dynasty ruler Vyagrahamukha. He was the son of Jishnugupta and was a Hindu by religion, in particular, a Shaivite . He lived and worked there for a good part of his life. Prithudaka Svamin , a later commentator, called him Bhillamalacharya , the teacher from Bhillamala. Bhillamala was the capital of the Gurjaradesa , the second-largest kingdom of Western India, comprising southern Rajasthan and northern Gujarat in modern-day India. It
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#17327659491091474-515: The Gregorian calendar (and its predecessor, the Julian calendar ), the world's most widely used calendar era . Common Era and Before the Common Era are alternatives to the original Anno Domini (AD) and Before Christ (BC) notations used for the same calendar era. The two notation systems are numerically equivalent: "2024 CE" and "AD 2024" each describe the current year; "400 BCE" and "400 BC" are
1541-599: The Gregorian calendar without the AD prefix. As early as 1825, the abbreviation VE (for Vulgar Era) was in use among Jews to denote years in the Western calendar. As of 2005 , Common Era notation has also been in use for Hebrew lessons for more than a century. Jews have also used the term Current Era . Some academics in the fields of theology , education , archaeology and history have adopted CE and BCE notation despite some disagreement. A study conducted in 2014 found that
1608-723: The National Trust said it would continue to use BC/AD as its house style. English Heritage explains its era policy thus: "It might seem strange to use a Christian calendar system when referring to British prehistory, but the BC/AD labels are widely used and understood." Some parts of the BBC use BCE/CE, but some presenters have said they will not. As of October 2019, the BBC News style guide has entries for AD and BC, but not for CE or BCE. The style guide for The Guardian says, under
1675-614: The date of birth of Jesus . Since the year numbers are the same, BCE and CE dates should be equally offensive to other religions as BC and AD. Roman Catholic priest and writer on interfaith issues Raimon Panikkar argued that the BCE/CE usage is the less inclusive option since they are still using the Christian calendar numbers and forcing it on other nations. In 1993, the English-language expert Kenneth G. Wilson speculated
1742-555: The 7th century Indian astronomical text by Brahmagupta , the Brāhmasphuṭasiddhānta , into Arabic as ' Zij as-Sindhind Az-Zīj ‛alā Sinī al-‛Arab , or the Sindhind . This translation was possibly the vehicle by means of which the mathematical methods of Indian astronomers were transmitted to Islam. The caliph ordered al-Fazārī to translate the Indian astronomical text, The Sindhind , along with Yaʿqūb ibn Ṭāriq , which
1809-587: The 8th and 9th centuries wrote commentaries on the Khanda-khadyaka . Further commentaries continued to be written into the 12th century. A few decades after the death of Brahmagupta, Sindh came under the Arab Caliphate in 712 CE. Expeditions were sent into Gurjaradesa (" Al-Baylaman in Jurz ", as per Arab historians). The kingdom of Bhillamala seems to have been annihilated but Ujjain repulsed
1876-596: The BCE/CE notation is not growing at the expense of BC and AD notation in the scholarly literature, and that both notations are used in a relatively stable fashion. In 2011, media reports suggested that the BC/AD notation in Australian school textbooks would be replaced by BCE/CE notation. The change drew opposition from some politicians and church leaders. Weeks after the story broke, the Australian Curriculum, Assessment and Reporting Authority denied
1943-579: The Christian Era, it was sometimes qualified, e.g., "common era of the Incarnation", "common era of the Nativity", or "common era of the birth of Christ". An adapted translation of Common Era into Latin as Era Vulgaris was adopted in the 20th century by some followers of Aleister Crowley , and thus the abbreviation "e.v." or "EV" may sometimes be seen as a replacement for AD. Although Jews have their own Hebrew calendar , they often use
2010-495: The Gregorian Calendar as BCE and CE without compromising their own beliefs about the divinity of Jesus of Nazareth." In History Today , Michael Ostling wrote: "BC/AD Dating: In the year of whose Lord? The continuing use of AD and BC is not only factually wrong but also offensive to many who are not Christians." Critics note the fact that there is no difference in the epoch of the two systems—chosen to be close to
2077-487: The Indians". In the Brāhmasphuṭasiddhānta , four methods for multiplication were described, including gomūtrikā , which is said to be close to the present day methods. In the beginning of chapter twelve of his Brāhmasphuṭasiddhānta , entitled "Calculation", he also details operations on fractions. The reader is expected to know the basic arithmetic operations as far as taking the square root, although he explains how to find
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2144-523: The Latin term anno aerae nostrae vulgaris may be that in a 1615 book by Johannes Kepler . Kepler uses it again, as ab Anno vulgaris aerae , in a 1616 table of ephemerides , and again, as ab anno vulgaris aerae , in 1617. A 1635 English edition of that book has the title page in English that may be the earliest-found use of Vulgar Era in English. A 1701 book edited by John Le Clerc includes
2211-625: The United States, the Kentucky State School Board reversed its decision to use BCE and CE in the state's new Program of Studies, leaving education of students about these concepts a matter of local discretion. The use of CE in Jewish scholarship was historically motivated by the desire to avoid the implicit "Our Lord" in the abbreviation AD . Although other aspects of dating systems are based in Christian origins, AD
2278-459: The attacks . The court of Caliph Al-Mansur (754–775) received an embassy from Sindh, including an astrologer called Kanaka, who brought (possibly memorised) astronomical texts, including those of Brahmagupta. Brahmagupta's texts were translated into Arabic by Muḥammad ibn Ibrāhīm al-Fazārī , an astronomer in Al-Mansur's court, under the names Sindhind and Arakhand . An immediate outcome was
2345-511: The column of the table in which he introduced the new era as " Anni Domini Nostri Jesu Christi " (Of the year of our Lord Jesus Christ]. This way of numbering years became more widespread in Europe with its use by Bede in England in 731. Bede also introduced the practice of dating years before what he supposed was the year of birth of Jesus, without a year zero . In 1422, Portugal became
2412-416: The constants c and e . The solution given is equivalent to x = e − c / b − d . He further gave two equivalent solutions to the general quadratic equation 18.44. Diminish by the middle [number] the square-root of the rupas multiplied by four times the square and increased by the square of the middle [number]; divide the remainder by twice the square. [The result is]
2479-413: The cube and cube-root of an integer and later gives rules facilitating the computation of squares and square roots. He then gives rules for dealing with five types of combinations of fractions: a / c + b / c ; a / c × b / d ; a / 1 + b / d ;
2546-489: The date that he believed to be the date of birth of Jesus , was conceived around the year 525 by the Christian monk Dionysius Exiguus . He did this to replace the then dominant Era of Martyrs system, because he did not wish to continue the memory of a tyrant who persecuted Christians. He numbered years from an initial reference date (" epoch "), an event he referred to as the Incarnation of Jesus. Dionysius labeled
2613-492: The desired variable must first be isolated, and then the equation must be divided by the desired variable's coefficient . In particular, he recommended using "the pulverizer" to solve equations with multiple unknowns. 18.51. Subtract the colors different from the first color. [The remainder] divided by the first [color's coefficient] is the measure of the first. [Terms] two by two [are] considered [when reduced to] similar divisors, [and so on] repeatedly. If there are many [colors],
2680-519: The early 20th century. The phrase "common era", in lower case , also appeared in the 19th century in a "generic" sense, not necessarily to refer to the Christian Era, but to any system of dates in common use throughout a civilization. Thus, "the common era of the Jews", "the common era of the Mahometans", "common era of the world", "the common era of the foundation of Rome". When it did refer to
2747-428: The entry for CE/BCE: "some people prefer CE (common era, current era, or Christian era) and BCE (before common era, etc.) to AD and BC, which, however, remain our style". In the United States, the use of the BCE/CE notation in textbooks was reported in 2005 to be growing. Some publications have transitioned to using it exclusively. For example, the 2007 World Almanac was the first edition to switch to BCE/CE, ending
Brahmagupta - Misplaced Pages Continue
2814-431: The first n natural numbers as n ( n + 1)(2 n + 1) / 6 and the sum of the cubes of the first n natural numbers as ( n ( n + 1) / 2 ) . Brahmagupta's Brahmasphuṭasiddhānta is the first book that provides rules for arithmetic manipulations that apply to zero and to negative numbers . The Brāhmasphuṭasiddhānta is the earliest known text to treat zero as
2881-453: The first of which was but eight days", and also refers to the common era as a synonym for vulgar era with "the fact that our Lord was born on the 4th year before the vulgar era, called Anno Domini, thus making (for example) the 42d year from his birth to correspond with the 38th of the common era". The Catholic Encyclopedia (1909) in at least one article reports all three terms (Christian, Vulgar, Common Era) being commonly understood by
2948-475: The following way: The multiplicand is repeated like a string for cattle, as often as there are integrant portions in the multiplier and is repeatedly multiplied by them and the products are added together. It is multiplication. Or the multiplicand is repeated as many times as there are component parts in the multiplier. Indian arithmetic was known in Medieval Europe as modus Indorum meaning "method of
3015-429: The general linear equation in chapter eighteen of Brahmasphuṭasiddhānta , The difference between rupas , when inverted and divided by the difference of the [coefficients] of the [unknowns], is the unknown in the equation. The rupas are [subtracted on the side] below that from which the square and the unknown are to be subtracted. which is a solution for the equation bx + c = dx + e where rupas refers to
3082-485: The grounds that BCE and CE are religiously neutral terms. They have been promoted as more sensitive to non-Christians by not referring to Jesus , the central figure of Christianity , especially via the religious terms " Christ " and Dominus ("Lord") used by the other abbreviations. Nevertheless, its epoch remains the same as that used for the Anno Domini era. The idea of numbering years beginning from
3149-459: The last Western European country to switch to the system begun by Dionysius. The term "Common Era" is traced back in English to its appearance as " Vulgar Era" to distinguish years of the Anno Domini era, which was in popular use, from dates of the regnal year (the year of the reign of a sovereign) typically used in national law. (The word 'vulgar' originally meant 'of the ordinary people', with no derogatory associations. ) The first use of
3216-456: The length, c , of its hypotenuse is given by c = m (1 + x ) − d . And, indeed, elementary algebraic manipulation shows that a + b = c whenever d has the value stated. Also, if m and x are rational, so are d , a , b and c . A Pythagorean triple can therefore be obtained from a , b and c by multiplying each of them by the least common multiple of their denominators . Brahmagupta went on to give
3283-436: The middle [number]. 18.45. Whatever is the square-root of the rupas multiplied by the square [and] increased by the square of half the unknown, diminished that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown. which are, respectively, solutions for the equation ax + bx = c equivalent to, and He went on to solve systems of simultaneous indeterminate equations stating that
3350-496: The phrase "Before Christ according to the Vulgar Æra, 6". The Merriam Webster Dictionary gives 1716 as the date of first use of the term "vulgar era" (which it defines as Christian era). The first published use of "Christian Era" may be the Latin phrase annus aerae christianae on the title page of a 1584 theology book, De Eucharistica controuersia . In 1649, the Latin phrase annus æræ Christianæ appeared in
3417-553: The phrase "before the common era" may be that in a 1770 work that also uses common era and vulgar era as synonyms, in a translation of a book originally written in German. The 1797 edition of the Encyclopædia Britannica uses the terms vulgar era and common era synonymously. In 1835, in his book Living Oracles , Alexander Campbell , wrote: "The vulgar Era, or Anno Domini; the fourth year of Jesus Christ,
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#17327659491093484-487: The pulverizer [is to be used]. Like the algebra of Diophantus , the algebra of Brahmagupta was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, similar to our notation but without the bar. Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms. The extent of Greek influence on this syncopation , if any,
3551-595: The rumours and stated that the BC/AD notation would remain, with CE and BCE as an optional suggested learning activity. In 2013, the Canadian Museum of Civilization (now the Canadian Museum of History) in Gatineau (opposite Ottawa ), which had previously switched to BCE/CE, decided to change back to BC/AD in material intended for the public while retaining BCE/CE in academic content. The notation
3618-540: The same year. The expression can be traced back to 1615, when it first appears in a book by Johannes Kepler as the Latin : annus aerae nostrae vulgaris ( year of our common era ), and to 1635 in English as " Vulgar Era". The term "Common Era" can be found in English as early as 1708, and became more widely used in the mid-19th century by Jewish religious scholars. Since the late 20th century, BCE and CE have become popular in academic and scientific publications on
3685-435: The sides diminished by [each] side of the quadrilateral. So given the lengths p , q , r and s of a cyclic quadrilateral, the approximate area is p + r / 2 · q + s / 2 while, letting t = p + q + r + s / 2 , the exact area is Common Era Common Era ( CE ) and Before the Common Era ( BCE ) are year notations for
3752-479: The spread of the decimal number system used in the texts. The mathematician Al-Khwarizmi (800–850 CE) wrote a text called al-Jam wal-tafriq bi hisal-al-Hind (Addition and Subtraction in Indian Arithmetic), which was translated into Latin in the 13th century as Algorithmi de numero indorum . Through these texts, the decimal number system and Brahmagupta's algorithms for arithmetic have spread throughout
3819-399: The squares is that [sum] multiplied by twice the [number of] step[s] increased by one [and] divided by three. The sum of the cubes is the square of that [sum] Piles of these with identical balls [can also be computed]. Here Brahmagupta found the result in terms of the sum of the first n integers, rather than in terms of n as is the modern practice. He gives the sum of the squares of
3886-637: The term "gurutvākarṣaṇam (गुरुत्वाकर्षणम्)" in Sanskrit to describe it. He is also credited with the first clear description of the quadratic formula (the solution of the quadratic equation) in his main work, the Brāhma-sphuṭa-siddhānta . Brahmagupta, according to his own statement, was born in 598 CE. Born in Bhillamāla in Gurjaradesa (modern Bhinmal in Rajasthan , India) during the reign of
3953-417: The title of an English almanac. A 1652 ephemeris may be the first instance found so far of the English use of "Christian Era". The English phrase "Common Era" appears at least as early as 1708, and in a 1715 book on astronomy it is used interchangeably with "Christian Era" and "Vulgar Era". A 1759 history book uses common æra in a generic sense, to refer to "the common era of the Jews". The first use of
4020-497: The traditional Jewish designations – B.C.E. and C.E. – cast a wider net of inclusion." In the World History Encyclopedia , Joshua J. Mark wrote "Non-Christian scholars, especially, embraced [CE and BCE] because they could now communicate more easily with the Christian community. Jewish, Islamic, Hindu and Buddhist scholars could retain their [own] calendar but refer to events using
4087-413: The world. Al-Khwarizmi also wrote his own version of Sindhind , drawing on Al-Fazari's version and incorporating Ptolemaic elements. Indian astronomic material circulated widely for centuries, even making its way into medieval Latin texts. The historian of science George Sarton called Brahmagupta "one of the greatest scientists of his race and the greatest of his time." Brahmagupta gave the solution of
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#17327659491094154-494: The year 628, at the age of 30, he composed the Brāhmasphuṭasiddhānta ("improved treatise of Brahma") which is believed to be a revised version of the received Siddhanta of the Brahmapaksha school of astronomy. Scholars state that he incorporated a great deal of originality into his revision, adding a considerable amount of new material. The book consists of 24 chapters with 1008 verses in the ārya metre . A good deal of it
4221-880: The year number. Unlike AD, which still often precedes the year number, CE always follows the year number (if context requires that it be written at all). Thus, the current year is written as 2024 in both notations (or, if further clarity is needed, as 2024 CE, or as AD 2024), and the year that Socrates died is represented as 399 BCE (the same year that is represented by 399 BC in the BC/AD notation). The abbreviations are sometimes written with small capital letters, or with periods (e.g., " B.C.E. " or "C.E."). The US-based Society of Biblical Literature style guide for academic texts on religion prefers BCE/CE to BC/AD. Mu%E1%B8%A5ammad ibn Ibr%C4%81h%C4%ABm al-Faz%C4%81r%C4%AB Muhammad ibn Ibrahim ibn Habib ibn Sulayman ibn Samra ibn Jundab al-Fazari ( Arabic : محمد بن إبراهيم بن حبيب بن سليمان بن سمرة بن جندب الفزاري ) (died 796 or 806)
4288-515: Was able to find integral solutions to Pell's equation through a series of equations of the form x − Ny = k i . Brahmagupta was not able to apply his solution uniformly for all possible values of N , rather he was only able to show that if x − Ny = k has an integer solution for k = ±1, ±2, or ±4, then x − Ny = 1 has a solution. The solution of the general Pell's equation would have to wait for Bhāskara II in c. 1150 CE . Brahmagupta's most famous result in geometry
4355-408: Was also a centre of learning for mathematics and astronomy. He became an astronomer of the Brahmapaksha school, one of the four major schools of Indian astronomy during this period. He studied the five traditional Siddhantas on Indian astronomy as well as the work of other astronomers including Aryabhata I , Latadeva, Pradyumna, Varahamihira , Simha, Srisena, Vijayanandin and Vishnuchandra. In
4422-628: Was an Arab philosopher , mathematician and astronomer . Al-Fazārī translated many scientific books into Arabic and Persian . He is credited to have built the first astrolabe in the Islamic world . He died in 796 or 806, possibly in Baghdad . At the end of the 8th century, whilst at the court of the Abbasid Caliphate , al-Fazārī mentioned Ghana , "the land of gold." Along with Yaʿqūb ibn Ṭāriq , al-Fazārī helped translate
4489-423: Was the identity, which is a generalisation of an identity that was discovered by Diophantus , Using his identity and the fact that if ( x 1 , y 1 ) and ( x 2 , y 2 ) are solutions to the equations x − Ny = k 1 and x − Ny = k 2 , respectively, then ( x 1 x 2 + Ny 1 y 2 , x 1 y 2 + x 2 y 1 ) is a solution to x − Ny = k 1 k 2 , he
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