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1. Brahmagupta
brahmagupta (598668). brahmagupta was born in 598 A.D.
##### Brahmagupta (598-668)
Brahmagupta was born in 598 A.D. in northwest India. He likely lived most of his life in Bhillamala (modern Bhinmal in Rajasthan) in the empire of Harsha. As a result Brahmagupta is often referred to as Bhillamalacarya, the teacher from Bhillamala. He belonged to the Ujjain school. Brahmagupta wrote his Brahma Sphuta Siddhanta at age 30. He gave his work this name since he brought up to date an old astronomical work, the Brahma Siddhanta. Brahmagupta's work is a compendious volume of astronomy. Four and a half chapters are devoted to pure math while his twelfth chapter, the Ganita, as the title reflects, deals with arithmetic, progressions and a bit of geometry. The eighteenth chapter of Brahmagupta's work is called the Kuttaka. Kuttaka generally means pulverizer. We usually associate the work Kuttaka with Aryabhata 's method for solving the indeterminate equation ax - by = c. But here Kuttaka means algebra (later Bija Ganita is used to connote algebra). Brahmagupta was the inventor of the concept of zero, the method of solving indeterminate equations of the second degree (ie. the solution of the equation Nx^2 + 1 = y^2 Bhaskara II was greatly influenced by Brahmagupta's work and gave Brahmagupta the title Ganita Chakra Chudamani, the gem of the circle of mathematicians.

2. Brahmagupta Polynomial -- From MathWorld
Astrónomo y matemático indio. Es, sin duda, el mayor matemático, de la antigua civilización india.
http://mathworld.wolfram.com/BrahmaguptaPolynomial.html
 Calculus and Analysis Special Functions Special Polynomials Brahmagupta Polynomial One of the polynomials obtained by taking powers of the Brahmagupta matrix . They satisfy the recurrence relation A list of many others is given by Suryanarayan (1996). Explicitly, The Brahmagupta polynomials satisfy The first few polynomials are and Taking and t = 2 gives equal to the Pell numbers and equal to half the Pell-Lucas numbers. The Brahmagupta polynomials are related to the Morgan-Voyce polynomials , but the relationship given by Suryanarayan (1996) is incorrect. Morgan-Voyce Polynomial References Suryanarayan, E. R. "The Brahmagupta Polynomials." Fib. Quart. Author: Eric W. Weisstein Wolfram Research, Inc.

 3. Encyclopædia Britannica brahmagupta was an orthodox Hindu, and his religious views, particularly the Hindu yuga system of measuring the ages ofhttp://www.britannica.com/eb/article?eu=16380

4. Quelques Grands Mathématiciens AC/BL Lycée Adam De Craponne Salon
SALON DE PROVENCE. brahmagupta. Né vers 598 à Multan, mort en 660 après JC.
http://pedagogie.ac-aix-marseille.fr/etablis/lycees/craponne/maths/brahma.htm
##### BRAHMAGUPTA
Né vers 598 à Multan, mort en 660 après JC.
Mathématicien et Astronome indien d'expression sanskrite.
Il était spécialiste en arithmétique, et a résolu des équations diophantiennes.
Il a utilisé le premier, le " zéro ".
Il a généralisé à un quadrilatère convexe inscriptible la formule de Héron
Si ABCD est un quadrilatère convexe inscriptible de côtés de longueurs a, b, c, d alors son aire est égale à : où Retour au tableau chronologique

5. Brahmagupta
brahmagupta. Born 598 brahmagupta, whose father was Jisnugupta, wroteimportant works on mathematics and astronomy. In particular
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Brahmagupta.html
##### Born: 598 in (possibly) Ujjain, India Died: 670 in India
Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index
Brahmagupta, whose father was Jisnugupta, wrote important works on mathematics and astronomy. In particular he wrote Brahmasphutasiddhanta (The Opening of the Universe), in 628. The work was written in 25 chapters and Brahmagupta tells us in the text that he wrote it at Bhillamala which today is the city of Bhinmal. This was the capital of the lands ruled by the Gurjara dynasty. Brahmagupta became the head of the astronomical observatory at Ujjain which was the foremost mathematical centre of ancient India at this time. Outstanding mathematicians such as Varahamihira had worked there and built up a strong school of mathematical astronomy. In addition to the Brahmasphutasiddhanta Brahmagupta wrote a second work on mathematics and astronomy which is the Khandakhadyaka written in 665 when he was 67 years old. We look below at some of the remarkable ideas which Brahmagupta's two treatises contain. First let us give an overview of their contents. The Brahmasphutasiddhanta contains twenty-five chapters but the first ten of these chapters seem to form what many historians believe was a first version of Brahmagupta's work and some manuscripts exist which contain only these chapters. These ten chapters are arranged in topics which are typical of Indian mathematical astronomy texts of the period. The topics covered are: mean longitudes of the planets; true longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; risings and settings; the moon's crescent; the moon's shadow;

6. Brahmagupta
Biography of brahmagupta, (598670) brahmagupta, whose father was Jisnugupta, wrote important works on mathematics and astronomy.
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Brahmagupta.html
##### Born: 598 in (possibly) Ujjain, India Died: 670 in India
Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index
Brahmagupta, whose father was Jisnugupta, wrote important works on mathematics and astronomy. In particular he wrote Brahmasphutasiddhanta (The Opening of the Universe), in 628. The work was written in 25 chapters and Brahmagupta tells us in the text that he wrote it at Bhillamala which today is the city of Bhinmal. This was the capital of the lands ruled by the Gurjara dynasty. Brahmagupta became the head of the astronomical observatory at Ujjain which was the foremost mathematical centre of ancient India at this time. Outstanding mathematicians such as Varahamihira had worked there and built up a strong school of mathematical astronomy. In addition to the Brahmasphutasiddhanta Brahmagupta wrote a second work on mathematics and astronomy which is the Khandakhadyaka written in 665 when he was 67 years old. We look below at some of the remarkable ideas which Brahmagupta's two treatises contain. First let us give an overview of their contents. The Brahmasphutasiddhanta contains twenty-five chapters but the first ten of these chapters seem to form what many historians believe was a first version of Brahmagupta's work and some manuscripts exist which contain only these chapters. These ten chapters are arranged in topics which are typical of Indian mathematical astronomy texts of the period. The topics covered are: mean longitudes of the planets; true longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; risings and settings; the moon's crescent; the moon's shadow;

7. References For Brahmagupta
References for brahmagupta,. Books HT Colebrooke, Algebra, with Arithmeticand Mensuration from the Sanscrit of brahmagupta and Bhaskara (1817).
http://www-gap.dcs.st-and.ac.uk/~history/References/Brahmagupta.html
##### References for Brahmagupta,
• Biography in Dictionary of Scientific Biography (New York 1970-1990).
• Biography in Encyclopaedia Britannica. Books:
• H T Colebrooke, Algebra, with Arithmetic and Mensuration from the Sanscrit of Brahmagupta and Bhaskara
• G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
• S S Prakash Sarasvati, A critical study of Brahmagupta and his works : The most distinguished Indian astronomer and mathematician of the sixth century A.D. (Delhi, 1986). Articles:
• S P Arya, On the Brahmagupta- Bhaskara equation, Math. Ed.
• G S Bhalla, Brahmagupta's quadrilateral, Math. Comput. Ed.
• B Chatterjee, Al-Biruni and Brahmagupta, Indian J. History Sci.
• B Datta, Brahmagupta, Bull. Calcutta Math. Soc.
• K Elfering, Die negativen Zahlen und die Rechenregeln mit ihnen bei Brahmagupta, in Mathemata, Boethius Texte Abh. Gesch. Exakt. Wissensch. XII (Wiesbaden, 1985, 83-86.
• R C Gupta, Brahmagupta's formulas for the area and diagonals of a cyclic quadrilateral, Math. Education
• 8. Varga Prakriti
The equation was nearly solved by brahmagupta, and was improved by Bhaskara(chakravala process). I followed the outline CN.Srinivasiengar
##### Varga Prakriti
Prakriti means coefficient, and refers to the coefficent N in the indeterminate equation Nx^2 + 1 = y^2, where N is a positive integer. By a mistake Euler refers to this equation as the Pell's equation, since he found the equation in an algebra text written by Pell. The equation was nearly solved by Brahmagupta , and was improved by Bhaskara chakravala process ). I followed the outline C.N.Srinivasiengar used in his: The History of Ancient Indian Mathematics, p110-111 to demonstrate Brahmagupta's solution to Nx^2 + 1 = y^2.
Brahmagupta calls N the "multiplier", x the "first root", and y the "last root". For conveniently chosen values of k and k', let (a,b), and (a',b') be a set of solutions of Nx^2 + k = y^2 and Nx^2 + k" = y^2. Then the principle of composition or Samasa (also known as Brahmagupta's Lemma) says that x = ab' + a'b, and y = bb' + Naa' are solutions of Nx^2 + kk' =y^2. If k = k', it follows that if Na^2 + k = b^2, then x = ab + ab = 2ab, y = bb + Naa = b^2 + Na^2 is a solution of Nx^2 + k^2 = y^2. Therefore, we have
N(2ab)^2 + k^2 =(b^2 + a^2N)^2. Dividing by k^2 we get

9. Brahmagupta
a topic from mathhistory-list brahmagupta post a message on this topic post a message on a new topic 2 Dec 1999 brahmagupta, by Heral Patel 2 Dec 1999 Re brahmagupta, by Sherman Stein 2 Dec 1999 Re brahmagupta, by David Wilkins
http://mathforum.com/epigone/math-history-list/phixquixclix
a topic from math-history-list
##### Brahmagupta
post a message on this topic
post a message on a new topic

2 Dec 1999 Brahmagupta , by Heral Patel
2 Dec 1999 Re: Brahmagupta , by Sherman Stein
2 Dec 1999 Re: Brahmagupta , by David Wilkins
2 Dec 1999 Re: Brahmagupta , by Randy K. Schwartz
3 Dec 1999 Re: Brahmagupta , by David Wilkins
The Math Forum

 10. BRAHMAGUPTA Translate this page brahmagupta (598-660). Astrónomo y matemático indio. Es, sin duda, elmayor matemático, de la antigua civilización india. Desarrollóhttp://almez.pntic.mec.es/~agos0000/Brahmagupta.html

 11. - Great Books - brahmagupta (598668), brahmagupta wrote important works on mathematicsand astronomy. In particular he wrote Brahmasphutasiddhantahttp://www.malaspina.com/site/person_240.asp?period_id=0&category_id=9

12. - Great Books -
brahmagupta (598668). brahmagupta wrote important works on mathematics and astronomy.
http://www.malaspina.com/site/person_240.asp
 Brahmagupta Brahmagupta wrote important works on mathematics and astronomy. In particular he wrote Brahmasphutasiddhanta (The Opening of the Universe), in 628. The work was written in 25 chapters and Brahmagupta tells us in the text that he wrote it at Bhillamala which today is the city of Bhinmal. This was the capital of the lands ruled by the Gurjara dynasty. Brahmagupta became the head of the astronomical observatory at Ujjain which was the foremost mathematical centre of ancient India at this time. Outstanding mathematicians such as Varahamihira had worked there and built up a strong school of mathematical astronomy. In addition to the Brahmasphutasiddhanta Brahmagupta wrote a second work on mathematics and astronomy which is the Khandakhadyaka written in 665 when he was 67 years old. [Adapted from MacTutor The Great Books Indian Mathematics This web page is part of a biographical database on Great Ideas . These are living ideas that have shaped, defined and directed world culture for over 2,500 years. The Great Ideas are radical, and often misunderstood and distorted by popular simplifications. Understanding a

13. Brahmagupta - Mathematics And The Liberal Arts
brahmagupta Mathematics and the Liberal Arts. The work of brahmaguptashould be relevant, but is not currently available in English.
http://math.truman.edu/~thammond/history/Brahmagupta.html
##### Brahmagupta - Mathematics and the Liberal Arts
To expand search, see India . Laterally related topics: The Hindu-Arabic Numerals Bhaskara Mahaviracarya Varahamihira ... The Tamil of South India , and The Sulvasutras The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews , published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet Biggs, N. L. The roots of combinatorics. Historia Math.

14. Brahmagupta's Formula For The Area Of A Cyclic Quadrilateral
brahmagupta's Formula. Problem Develop a proof for brahmagupta'sFormula. Who was brahmagupta? brahmagupta's formula is provides
http://jwilson.coe.uga.edu/emt725/brahmagupta/brahmagupta.html
##### Problem: Develop a proof for Brahmagupta's Formula.
Who was Brahmagupta?
Brahmagupta's formula is provides the area A of a cyclic quadrilateral (i.e., a simple quadrilateral that is inscribed in a circle) with sides of length a, b, c, and d as
where s is the semiperimeter
Note: There are alternative approaches to this proof. The one outlined below is intuitive and elementary, but becomes tedious. A more elegant approach is available using trigonometry.
The use of Ptolemy's theorem (the product of the diagonals equals the sum of the products of opposite sides) may provide a different investigation of the problem.
If ABCD is a rectangle the formula is clear.
Consider the chord AC.
The angle that subtends a chord has measure that is half the measure of the intercepted arc. But the chord AC is simultaneously subtended by the angle at B and by the angle at D. There for the sum of these angles is 180 degrees. Opposite angles of a cyclic quadrilateral are supplemental.
Assume the quadrilateral is not a rectangle. WNLOG, extend AB and CD until they meet at P.
Label the extensions outside the circle e and f.

15. Brahmagupta's Formula
brahmagupta's Formula. by. Kala Fischbein and Tammy Brooks. brahmagupta'sFormula. Prove For a cyclic quadrilateral with sides
http://jwilson.coe.uga.edu/emt725/Class/Brooks/Brahmagupta/Brahmagupta.html
##### Brahmagupta's Formula
Prove: For a cyclic quadrilateral with sides of length a, b, c, and d, the area is given by
where s is the semiperimeter.
Given:
Draw chord AC. Extend AB and CD so they meet at point P.
Angle ADC and Angle ABC subtend the same chord AC from the two arcs of the circle. Therefore they are supplementary. Angle ADP is supplementary to Angle ADC. So
Triangle PBC and Triangle PDA are similar. The ratio of similarity is
Area of Triangle PDA = * Area of Triangle PBC
Area ABCD = Area of Triangle PBC - Area of Triangle PDA
Let A = Area of quadrilateral ABCD and T = Area of Triangle PBC.
A = T - T = T = T Let PA = e and PD = f. Applying Heron's Formula, the area of triangle PBC is Therefore, (Note: We have used s at this point for the semiperimeter of the TRIANGLE. In what follows, we will substitute for s, e, and f in terms of a, b, c, and d. Eventually we will return to the use of s to represent the semiperimeter of the quadrilateral. 1. First, we want a substitution for e in terms of a, b, c, and d.

16. References For Brahmagupta
References for the biography of brahmagupta, S P Arya, On the brahmagupta Bhaskara equation, Math.
http://www-groups.dcs.st-and.ac.uk/~history/References/Brahmagupta.html
##### References for Brahmagupta,
• Biography in Dictionary of Scientific Biography (New York 1970-1990).
• Biography in Encyclopaedia Britannica. Books:
• H T Colebrooke, Algebra, with Arithmetic and Mensuration from the Sanscrit of Brahmagupta and Bhaskara
• G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
• S S Prakash Sarasvati, A critical study of Brahmagupta and his works : The most distinguished Indian astronomer and mathematician of the sixth century A.D. (Delhi, 1986). Articles:
• S P Arya, On the Brahmagupta- Bhaskara equation, Math. Ed.
• G S Bhalla, Brahmagupta's quadrilateral, Math. Comput. Ed.
• B Chatterjee, Al-Biruni and Brahmagupta, Indian J. History Sci.
• B Datta, Brahmagupta, Bull. Calcutta Math. Soc.
• K Elfering, Die negativen Zahlen und die Rechenregeln mit ihnen bei Brahmagupta, in Mathemata, Boethius Texte Abh. Gesch. Exakt. Wissensch. XII (Wiesbaden, 1985, 83-86.
• R C Gupta, Brahmagupta's formulas for the area and diagonals of a cyclic quadrilateral, Math. Education
• 17. Brahmagupta
Translate this page brahmagupta indien, 598-660 (?) Son apparition en Inde, tout particulièrementdans l'oeuvre de brahmagupta, est un pas de géant en algèbre.
http://www.sciences-en-ligne.com/momo/chronomath/chrono1/Brahmagupta.html
 BRAHMAGUPTA indien, 598-660 indien Aryabhata Chuquet : Arabes Al- Khwarizmi babyloniennes et grecques . Son apparition en Inde Diophante et Aryabhata x = ny (lorsque n est entier) Pell Bhaskara Notons que le concept de congruence Gauss Brahamagupta dans son Bhrama Sphuta Siddhanta n et n chinoises lunaisons Dans son livre Victor J. Katz cite un cas n Gauss et les congruences : inscriptible A Pour en savoir plus : A HISTORY OF MATHEMATICS , an introduction, par Victor J. KATZ Addison-Wesley Educational Publishers -1999 Boece Khwarizmi

18. Brahmagupta's Problem -- From MathWorld
MathWorld Logo. Alphabetical Index. Eric's other sites. Number Theory , DiophantineEquations v. brahmagupta's Problem, Solve the Pell equation in integers.
http://mathworld.wolfram.com/BrahmaguptasProblem.html
 Number Theory Diophantine Equations Brahmagupta's Problem Solve the Pell equation in integers . The smallest solution is x y Diophantine Equation Pell Equation Author: Eric W. Weisstein Wolfram Research, Inc.

 19. Brahmagupta brahmagupta, brä mugoop'tu Pronunciation Key. brahmagupta , c. 598c. 660, Indianmathematician and astronomer. . from the Sanskrit of brahmagupta (1817).http://www.infoplease.com/ce5/CE007231.html

20. Brahmagupta's Theorem
brahmagupta's Theorem In a cyclic quadrilateral having perpendicular diagonals,the perpendicular to a side from the point of intersection of the diagonals
http://www.cut-the-knot.com/Curriculum/Geometry/Brahmagupta.shtml
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##### Brahmagupta's Theorem: What is it? A Mathematical Droodle
Explanation Alexander Bogomolny
##### Brahmagupta 's Theorem
In a cyclic quadrilateral having perpendicular diagonals, the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side. There are several right angles: DET in EDT, AET in AET, DTA in ADT, BTC in BCT. From the first three triangles, we have DTE = EAT and ETA = EDT. We also have two pairs of vertically opposite angles: DTE = BTQ and ETA = CTQ. Chords AB and DC subtend pairs of angles: ADB = ACB and DAC = DBC. By comparing (1)-(3) we conclude that TCQ = CTQ and BTQ = TBQ. Both triangles CQT and BQT are isosceles and BQ = QT = CQ. The theorem of course admits the following variation: In a cyclic quadrilateral having perpendicular diagonals, the perpendicular from the midpoint of a side to the opposite side passes through the point of intersection of the diagonals. There are four such perpendiculars and all four pass through the point of intersection of the diagonals. In other words, the four perpendiculars from the midpoints of the sides to the opposite side are concurrent, and the point of concurrency coincides with the intersection of the diagonals. Now, all this is true under the condition of orthogonality of the diagonals. Orthogonality plays an important role in both the formulation and the proof of the theorem. It's therefore a curiosity that the theorem admits a generalization that does not require the diagonals to be orthogonal. In the

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