Английская Википедия:Al-Khwarizmi
Шаблон:Short description Шаблон:Pp-semi Шаблон:Pp-move-indef Шаблон:Other uses Шаблон:Infobox academic Шаблон:Use dmy dates Шаблон:Use Oxford spelling
Muhammad ibn Musa al-KhwarizmiШаблон:Refn (Шаблон:Lang-ar; Шаблон:Circa), often referred to as simply al-Khwarizmi, was a Persian polymath who produced vastly influential Arabic-language works in mathematics, astronomy, and geography. Hailing from Khwarazm, he was appointed as the astronomer and head of the House of Wisdom in the city of Baghdad around 820 CE.
His popularizing treatise on algebra, compiled between 813-33 as Al-Jabr (The Compendious Book on Calculation by Completion and Balancing),[1]Шаблон:Rp presented the first systematic solution of linear and quadratic equations. One of his achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications.[2]Шаблон:Rp Because al-Khwarizmi was the first person to treat algebra as an independent discipline and introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation),[3] he has been described as the father[4][5][6] or founder[7][8] of algebra. The English term algebra comes from the short-hand title of his aforementioned treatise (Шаблон:Lang Шаблон:Transliteration, Шаблон:Translation).[9] His name gave rise to the English terms algorism and algorithm; the Spanish, Italian, and Portuguese terms Шаблон:Text; and the Spanish term Шаблон:Lang[10] and Portuguese term Шаблон:Lang, both meaning "digit".[11]
In the 12th century, Latin-language translations of al-Khwarizmi's textbook on Indian arithmetic (Шаблон:Lang-la), which codified the various Indian numerals, introduced the decimal-based positional number system to the Western world.[12] Likewise, Al-Jabr, translated into Latin by the English scholar Robert of Chester in 1145, was used until the 16th century as the principal mathematical textbook of European universities.[13][14][15][16]
Al-Khwarizmi revised Geography, the 2nd-century Greek-language treatise by the Roman polymath Claudius Ptolemy, listing the longitudes and latitudes of cities and localities.[17]Шаблон:Rp He further produced a set of astronomical tables and wrote about calendric works, as well as the astrolabe and the sundial.[18] Al-Khwarizmi made important contributions to trigonometry, producing accurate sine and cosine tables and the first table of tangents.
Life
Few details of al-Khwārizmī's life are known with certainty. Ibn al-Nadim gives his birthplace as Khwarazm, and he is generally thought to have come from this region.[19][20][21] Of Persian stock,[22][19][23][24][25] his name means 'the native of Khwarazm', a region that was part of Greater Iran,[26] and is now part of Turkmenistan, and Uzbekistan.[27]
Al-Tabari gives his name as Muḥammad ibn Musá al-Khwārizmī al-Majūsī al-Quṭrubbullī (Шаблон:Lang). The epithet al-Qutrubbulli could indicate he might instead have come from Qutrubbul (Qatrabbul),[28] near Baghdad. However, Roshdi Rashed denies this:[29] Шаблон:Blockquote
On the other hand, David A. King affirms his nisba to Qutrubul, noting that he was called al-Khwārizmī al-Qutrubbulli because he was born just outside of Baghdad.[30]
Regarding al-Khwārizmī's religion, Toomer writes:[31]
Ibn al-Nadīm's Шаблон:Lang includes a short biography on al-Khwārizmī together with a list of his books. Al-Khwārizmī accomplished most of his work between 813 and 833. After the Muslim conquest of Persia, Baghdad had become the centre of scientific studies and trade. Around 820 CE, he was appointed as the astronomer and head of the library of the House of Wisdom.[2]Шаблон:Rp The House of Wisdom was established by the Abbasid Caliph al-Ma'mūn. Al-Khwārizmī studied sciences and mathematics, including the translation of Greek and Sanskrit scientific manuscripts. He was also a historian who is cited by the likes of al-Tabari and Ibn Abi Tahir.[32]
During the reign of al-Wathiq, he is said to have been involved in the first of two embassies to the Khazars.[33] Douglas Morton Dunlop suggests that Muḥammad ibn Mūsā al-Khwārizmī might have been the same person as Muḥammad ibn Mūsā ibn Shākir, the eldest of the three Banū Mūsā brothers.[34]
Contributions
Al-Khwārizmī's contributions to mathematics, geography, astronomy, and cartography established the basis for innovation in algebra and trigonometry. His systematic approach to solving linear and quadratic equations led to algebra, a word derived from the title of his book on the subject, Al-Jabr.[35]
On the Calculation with Hindu Numerals, written about 820, was principally responsible for spreading the Hindu–Arabic numeral system throughout the Middle East and Europe. It was translated into Latin as Algoritmi de numero Indorum. Al-Khwārizmī, rendered in Latin as Algoritmi, led to the term "algorithm".[36][37]
Some of his work was based on Persian and Babylonian astronomy, Indian numbers, and Greek mathematics.
Al-Khwārizmī systematized and corrected Ptolemy's data for Africa and the Middle East. Another major book was Kitab surat al-ard ("The Image of the Earth"; translated as Geography), presenting the coordinates of places based on those in the Geography of Ptolemy, but with improved values for the Mediterranean Sea, Asia, and Africa.[38]
He wrote on mechanical devices like the astrolabe[39] and sundial.[18] He assisted a project to determine the circumference of the Earth and in making a world map for al-Ma'mun, the caliph, overseeing 70 geographers.[40] When, in the 12th century, his works spread to Europe through Latin translations, it had a profound impact on the advance of mathematics in Europe.[41]
Algebra
Шаблон:Main Шаблон:Further Шаблон:Multiple image
Al-Jabr (The Compendious Book on Calculation by Completion and Balancing, Шаблон:Lang-ar Шаблон:Transliteration) is a mathematical book written approximately 820 CE. It was written with the encouragement of Caliph al-Ma'mun as a popular work on calculation and is replete with examples and applications to a range of problems in trade, surveying and legal inheritance.[42] The term "algebra" is derived from the name of one of the basic operations with equations (Шаблон:Transliteration, meaning "restoration", referring to adding a number to both sides of the equation to consolidate or cancel terms) described in this book. The book was translated in Latin as Liber algebrae et almucabala by Robert of Chester (Segovia, 1145) hence "algebra", and by Gerard of Cremona. A unique Arabic copy is kept at Oxford and was translated in 1831 by F. Rosen. A Latin translation is kept in Cambridge.[43]
It provided an exhaustive account of solving polynomial equations up to the second degree,Шаблон:Sfn and discussed the fundamental method of "reduction" and "balancing", referring to the transposition of terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation.[44]
Al-Khwārizmī's method of solving linear and quadratic equations worked by first reducing the equation to one of six standard forms (where b and c are positive integers)
- squares equal roots (ax2 = bx)
- squares equal number (ax2 = c)
- roots equal number (bx = c)
- squares and roots equal number (ax2 + bx = c)
- squares and number equal roots (ax2 + c = bx)
- roots and number equal squares (bx + c = ax2)
by dividing out the coefficient of the square and using the two operations Шаблон:Transliteration (Шаблон:Lang-ar "restoring" or "completion") and Шаблон:Transliteration ("balancing"). Шаблон:Transliteration is the process of removing negative units, roots and squares from the equation by adding the same quantity to each side. For example, x2 = 40x − 4x2 is reduced to 5x2 = 40x. Шаблон:Transliteration is the process of bringing quantities of the same type to the same side of the equation. For example, x2 + 14 = x + 5 is reduced to x2 + 9 = x.
The above discussion uses modern mathematical notation for the types of problems that the book discusses. However, in al-Khwārizmī's day, most of this notation had not yet been invented, so he had to use ordinary text to present problems and their solutions. For example, for one problem he writes, (from an 1831 translation) Шаблон:Blockquote In modern notation this process, with x the "thing" (Шаблон:Lang shayʾ) or "root", is given by the steps,
- <math>(10-x)^2=81 x</math>
- <math>100 + x^2 - 20 x = 81 x</math>
- <math>x^2+100=101 x</math>
Let the roots of the equation be x = p and x = q. Then <math>\tfrac{p+q}{2}=50\tfrac{1}{2}</math>, <math>pq =100</math> and
- <math>\frac{p-q}{2} = \sqrt{\left(\frac{p+q}{2}\right)^2 - pq}=\sqrt{2550\tfrac{1}{4} - 100}=49\tfrac{1}{2}</math>
So a root is given by
- <math>x=50\tfrac{1}{2}-49\tfrac{1}{2}=1</math>
Several authors have published texts under the name of Шаблон:Transliteration, including Abū Ḥanīfa Dīnawarī, Abū Kāmil, Abū Muḥammad al-'Adlī, Abū Yūsuf al-Miṣṣīṣī, 'Abd al-Hamīd ibn Turk, Sind ibn 'Alī, Sahl ibn Bišr, and Sharaf al-Dīn al-Ṭūsī.
Solomon Gandz has described Al-Khwarizmi as the father of Algebra: Шаблон:Blockquote
Victor J. Katz adds : Шаблон:Blockquote
John J. O'Connor and Edmund F. Robertson wrote in the MacTutor History of Mathematics Archive: Шаблон:Blockquote
Roshdi Rashed and Angela Armstrong write: Шаблон:Blockquote
According to Swiss-American historian of mathematics, Florian Cajori, Al-Khwarizmi's algebra was different from the work of Indian mathematicians, for Indians had no rules like the ''restoration'' and ''reduction''.[45] Regarding the dissimilarity and significance of Al-Khwarizmi's algebraic work from that of Indian Mathematician Brahmagupta, Carl B. Boyer wrote:
It is true that in two respects the work of al-Khowarizmi represented a retrogression from that of Diophantus. First, it is on a far more elementary level than that found in the Diophantine problems and, second, the algebra of al-Khowarizmi is thoroughly rhetorical, with none of the syncopation found in the Greek Arithmetica or in Brahmagupta's work. Even numbers were written out in words rather than symbols! It is quite unlikely that al-Khwarizmi knew of the work of Diophantus, but he must have been familiar with at least the astronomical and computational portions of Brahmagupta; yet neither al-Khwarizmi nor other Arabic scholars made use of syncopation or of negative numbers. Nevertheless, the Al-jabr comes closer to the elementary algebra of today than the works of either Diophantus or Brahmagupta, because the book is not concerned with difficult problems in indeterminant analysis but with a straight forward and elementary exposition of the solution of equations, especially that of second degree. The Arabs in general loved a good clear argument from premise to conclusion, as well as systematic organization – respects in which neither Diophantus nor the Hindus excelled.[46]
Arithmetic
Al-Khwārizmī's second most influential work was on the subject of arithmetic, which survived in Latin translations but is lost in the original Arabic. His writings include the text kitāb al-ḥisāb al-hindī ('Book of Indian computation'Шаблон:Refn), and perhaps a more elementary text, kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī ('Addition and subtraction in Indian arithmetic').Шаблон:Sfn[47] These texts described algorithms on decimal numbers (Hindu–Arabic numerals) that could be carried out on a dust board. Called takht in Arabic (Latin: tabula), a board covered with a thin layer of dust or sand was employed for calculations, on which figures could be written with a stylus and easily erased and replaced when necessary. Al-Khwarizmi's algorithms were used for almost three centuries, until replaced by Al-Uqlidisi's algorithms that could be carried out with pen and paper.[48]
As part of 12th century wave of Arabic science flowing into Europe via translations, these texts proved to be revolutionary in Europe.[49] Al-Khwarizmi's Latinized name, Algorismus, turned into the name of method used for computations, and survives in the term "algorithm". It gradually replaced the previous abacus-based methods used in Europe.[50]
Four Latin texts providing adaptions of Al-Khwarizmi's methods have survived, even though none of them is believed to be a literal translation:Шаблон:Sfn
- Dixit Algorizmi (published in 1857 under the title Algoritmi de Numero Indorum[51])[52]
- Liber Alchoarismi de Practica Arismetice
- Liber Ysagogarum Alchorismi
- Liber Pulveris
Dixit Algorizmi ('Thus spake Al-Khwarizmi') is the starting phrase of a manuscript in the University of Cambridge library, which is generally referred to by its 1857 title Algoritmi de Numero Indorum. It is attributed to the Adelard of Bath, who had translated the astronomical tables in 1126. It is perhaps the closest to Al-Khwarizmi's own writings.[52]
Al-Khwarizmi's work on arithmetic was responsible for introducing the Arabic numerals, based on the Hindu–Arabic numeral system developed in Indian mathematics, to the Western world. The term "algorithm" is derived from the algorism, the technique of performing arithmetic with Hindu-Arabic numerals developed by al-Khwārizmī. Both "algorithm" and "algorism" are derived from the Latinized forms of al-Khwārizmī's name, Algoritmi and Algorismi, respectively.[53]
Astronomy
Al-Khwārizmī's [[Zij as-Sindhind|Шаблон:Transliteration]][31] (Шаблон:Lang-ar, "astronomical tables of Siddhanta"[54]) is a work consisting of approximately 37 chapters on calendrical and astronomical calculations and 116 tables with calendrical, astronomical and astrological data, as well as a table of sine values. This is the first of many Arabic Zijes based on the Indian astronomical methods known as the sindhind.[55] The word Sindhind is a corruption of the Sanskrit Siddhānta, which is the usual designation of an astronomical textbook. In fact, the mean motions in the tables of al-Khwarizmi are derived from those in the "corrected Brahmasiddhanta" (Brahmasphutasiddhanta) of Brahmagupta.[56]
The work contains tables for the movements of the sun, the moon and the five planets known at the time. This work marked the turning point in Islamic astronomy. Hitherto, Muslim astronomers had adopted a primarily research approach to the field, translating works of others and learning already discovered knowledge.
The original Arabic version (written Шаблон:Circa) is lost, but a version by the Spanish astronomer Maslama al-Majriti (Шаблон:Circa) has survived in a Latin translation, presumably by Adelard of Bath (26 January 1126).[57] The four surviving manuscripts of the Latin translation are kept at the Bibliothèque publique (Chartres), the Bibliothèque Mazarine (Paris), the Biblioteca Nacional (Madrid) and the Bodleian Library (Oxford).
Trigonometry
Al-Khwārizmī's Zīj as-Sindhind contained tables for the trigonometric functions of sines and cosine.[55] A related treatise on spherical trigonometry is attributed to him.[58]
Al-Khwārizmī produced accurate sine and cosine tables, and the first table of tangents.[59][60]
Geography
Al-Khwārizmī's third major work is his Шаблон:Transliteration (Шаблон:Lang-ar, "Book of the Description of the Earth"),Шаблон:Refn also known as his Geography, which was finished in 833. It is a major reworking of Ptolemy's second-century Geography, consisting of a list of 2402 coordinates of cities and other geographical features following a general introduction.[61]
There is one surviving copy of Шаблон:Transliteration, which is kept at the Strasbourg University Library. A Latin translation is at the Biblioteca Nacional de España in Madrid.[62] The book opens with the list of latitudes and longitudes, in order of "weather zones", that is to say in blocks of latitudes and, in each weather zone, by order of longitude. As Paul Gallez notes, this system allows the deduction of many latitudes and longitudes where the only extant document is in such a bad condition, as to make it practically illegible. Neither the Arabic copy nor the Latin translation include the map of the world; however, Hubert Daunicht was able to reconstruct the missing map from the list of coordinates. Daunicht read the latitudes and longitudes of the coastal points in the manuscript, or deduced them from the context where they were not legible. He transferred the points onto graph paper and connected them with straight lines, obtaining an approximation of the coastline as it was on the original map. He did the same for the rivers and towns.[63]
Al-Khwārizmī corrected Ptolemy's gross overestimate for the length of the Mediterranean Sea[64] from the Canary Islands to the eastern shores of the Mediterranean; Ptolemy overestimated it at 63 degrees of longitude, while al-Khwārizmī almost correctly estimated it at nearly 50 degrees of longitude. He "depicted the Atlantic and Indian Oceans as open bodies of water, not land-locked seas as Ptolemy had done."[65] Al-Khwārizmī's Prime Meridian at the Fortunate Isles was thus around 10° east of the line used by Marinus and Ptolemy. Most medieval Muslim gazetteers continued to use al-Khwārizmī's prime meridian.[64]
Jewish calendar
Al-Khwārizmī wrote several other works including a treatise on the Hebrew calendar, titled Шаблон:Transliteration (Шаблон:Lang-ar, "Extraction of the Jewish Era"). It describes the Metonic cycle, a 19-year intercalation cycle; the rules for determining on what day of the week the first day of the month Tishrei shall fall; calculates the interval between the Anno Mundi or Jewish year and the Seleucid era; and gives rules for determining the mean longitude of the sun and the moon using the Hebrew calendar. Similar material is found in the works of Al-Bīrūnī and Maimonides.[31]
Other works
Ibn al-Nadim's Шаблон:Transliteration, an index of Arabic books, mentions al-Khwārizmī's Шаблон:Transliteration (Шаблон:Lang-ar), a book of annals. No direct manuscript survives; however, a copy had reached Nusaybin by the 11th century, where its metropolitan bishop, Mar Elias bar Shinaya, found it. Elias's chronicle quotes it from "the death of the Prophet" through to 169 AH, at which point Elias's text itself hits a lacuna.[66]
Several Arabic manuscripts in Berlin, Istanbul, Tashkent, Cairo and Paris contain further material that surely or with some probability comes from al-Khwārizmī. The Istanbul manuscript contains a paper on sundials; the Fihrist credits al-Khwārizmī with Шаблон:Transliteration (Шаблон:Lang-ar). Other papers, such as one on the determination of the direction of Mecca, are on the spherical astronomy.
Two texts deserve special interest on the morning width (Шаблон:Transliteration) and the determination of the azimuth from a height (Шаблон:Transliteration). He wrote two books on using and constructing astrolabes.
Honours
- Al-Khwarizmi (crater) — A crater on the far side of the Moon. [67]
- 13498 Al Chwarizmi — Main-belt Asteroid, Discovered 1986 Aug 6 by E. W. Elst and V. G. Ivanova at Smolyan.[68]
- 11156 Al-Khwarismi — Main-belt Asteroid, Discovered 1997 Dec 31 by P. G. Comba at Prescott.[69]
Notes
References
Sources
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Citation
- Шаблон:Cite book
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Citation
- Шаблон:Cite book
- Шаблон:Cite encyclopedia
Further reading
Biographical
- Brentjes, Sonja (2007). "Khwārizmī: Muḥammad ibn Mūsā al‐Khwārizmī Шаблон:Webarchive" in Thomas Hockey et al.(eds.). The Biographical Encyclopedia of Astronomers, Springer Reference. New York: Springer, 2007, pp. 631–633. (PDF version Шаблон:Webarchive)
- Hogendijk, Jan P., Muhammad ibn Musa (Al-)Khwarizmi (c. 780–850 CE) Шаблон:Webarchive – bibliography of his works, manuscripts, editions and translations.
- Шаблон:MacTutor Biography
- Sezgin, F., ed., Islamic Mathematics and Astronomy, Frankfurt: Institut für Geschichte der arabisch-islamischen Wissenschaften, 1997–99.
Algebra
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Hughes, Barnabas. Robert of Chester's Latin translation of al-Khwarizmi's al-Jabr: A new critical edition. In Latin. F. Steiner Verlag Wiesbaden (1989). Шаблон:Isbn.
- Шаблон:Cite book
- Шаблон:Cite book
Astronomy
- Шаблон:Cite book
- Шаблон:Cite journal (Hogendijk's homepage. Publication in English, no. 25).
- Шаблон:Cite book (Description and analysis of seven recently discovered minor works related to al-Khwarizmi).
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book (Van Dalen's homepage. List of Publications, Articles – no. 5).
Jewish calendar
External links
Шаблон:Navboxes Шаблон:Portal bar Шаблон:Authority control
- ↑ Oaks, J. (2009), "Polynomials and Equations in Arabic Algebra", Archive for History of Exact Sciences, 63(2), 169–203.
- ↑ 2,0 2,1 Maher, P. (1998), "From Al-Jabr to Algebra", Mathematics in School, 27(4), 14–15.
- ↑ (Boyer 1991, "The Arabic Hegemony" p. 229) "It is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the translation above. The word al-jabr presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word muqabalah is said to refer to "reduction" or "balancing" – that is, the cancellation of like terms on opposite sides of the equation."
- ↑ Шаблон:Cite book
- ↑ Boyer, Carl B., 1985. A History of Mathematics, p. 252. Princeton University Press. "Diophantus sometimes is called the father of algebra, but this title more appropriately belongs to al-Khowarizmi...", "...the Al-jabr comes closer to the elementary algebra of today than the works of either Diophantus or Brahmagupta..."
- ↑ Gandz, Solomon, The sources of al-Khwarizmi's algebra, Osiris, i (1936), 263–277, "Al-Khwarizmi's algebra is regarded as the foundation and cornerstone of the sciences. In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers."
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Harvnb
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ van der Waerden, Bartel Leendert (1985). A History of Algebra: From al–Khwarizmi to Emmy Noether. Berlin: Springer-Verlag.
- ↑ 18,0 18,1 Шаблон:Harvnb
- ↑ 19,0 19,1 Шаблон:Cite journal
- ↑ Шаблон:Cite book
"Ibn al-Nadīm and Ibn al-Qifṭī relate that al-Khwārizmī's family came from Khwārizm, the region south of the Aral sea."
Also → al-Nadīm, Abu'l-Faraj (1871–1872). Kitāb al-Fihrist, ed. Gustav Flügel, Leipzig: Vogel, p. 274. al-Qifṭī, Jamāl al-Dīn (1903). Taʾrīkh al-Hukamā, eds. August Müller & Julius Lippert, Leipzig: Theodor Weicher, p. 286. - ↑ Шаблон:Citation
- ↑ Шаблон:Cite book
- ↑ A History of Science in World Cultures: Voices of Knowledge. Routledge. Page 228. "Mohammed ibn Musa al-Khwarizmi (780–850) was a Persian astronomer and mathematician from the district of Khwarism (Uzbekistan area of Central Asia)."
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Encycloaedia Iranica-online, s.v. "CHORASMIA, ii. In Islamic times Шаблон:Webarchive," by Clifford E. Bosworth.
- ↑ Шаблон:Cite book
- ↑ "Iraq After the Muslim Conquest", by Michael G. Morony, Шаблон:Isbn (a 2005 facsimile from the original 1984 book), p. 145 Шаблон:Webarchive
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite AV media
- ↑ 31,0 31,1 31,2 Шаблон:Harvnb
- ↑ Шаблон:The History of al-Tabari
- ↑ Шаблон:Cite book
- ↑ Шаблон:Harvnb
- ↑ Шаблон:Cite web
- ↑ Шаблон:Harvnb
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite web
- ↑ Joseph Frank, al-Khwarizmi über das Astrolab, 1922.
- ↑ Шаблон:Cite encyclopedia
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Harv "It is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the translation above. The word al-jabr presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word muqabalah is said to refer to "reduction" or "balancing" — that is, the cancellation of like terms on opposite sides of the equation."
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Citation
- ↑ Шаблон:Citation
- ↑ Шаблон:Citation
- ↑ Шаблон:Citation
- ↑ Шаблон:Citation
- ↑ 52,0 52,1 Шаблон:Citation
- ↑ Шаблон:Cite web
- ↑ Шаблон:Citation
- ↑ 55,0 55,1 Шаблон:Harvnb
- ↑ Шаблон:Cite book
- ↑ Шаблон:Harvnb
- ↑ Ошибка цитирования Неверный тег
<ref>; для сносокMacTutorне указан текст - ↑ Jacques Sesiano, "Islamic mathematics", p. 157, in Шаблон:Cite book
- ↑ Шаблон:Cite encyclopedia
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite book
- ↑ Daunicht
- ↑ 64,0 64,1 Edward S. Kennedy, Mathematical Geography, p. 188, in Шаблон:Harv
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite journal NASA Portal: Apollo 11, Photography Index.
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
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