Discover

'Brahmagupta (ब्रह्मगुप्त)' () (589–668) was an Indian mathematician and astronomer. Brahmagupta was born in 598 A.D. in Bhinmal city in the state of Rajasthan of 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 Bhinmal.
He was the head of the astronomical observatory at Ujjain, and during his tenure there wrote four texts on mathematics and astronomy: the ''Brahmasphutasiddhanta'' in 628, the ''Cadamekela'' in 624, the ''Durkeamynarda'' in 672, and the ''Khandakhadyaka'' in 665.

Contents

Mathematics

Astronomy

Notes

Mathematics

Addition, subtraction, division and other fundamental operations using Hindu Arabic numerals first appear in Brahmasputha Siddhanta which was translated to Arabic as "Sindhind". Brahmagupta is known to have passed on knowledge to his students "orally" through expressions. He also tended to omit words and steps so that his mathematical works could be presented in a more poetic manner. For instance, Brahmagupta gives the sum of the squares of the first n natural numbers as n(n+1)(2n+1)/6 and the sum of the cubes of the first n natural numbers as (n(n+1)/2)².As such no proofs are given as to how Brahmagupta discovered these formulae.[1]
Nevertheless, his work has made a significant impact in mathematical constructs. Brahmagupta popularized an important concept in mathematics: the number zero. The ''Brahmasphutasiddhanta'' is the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number. It goes well beyond that, however, stating rules for arithmetic on negative numbers and zero which are quite close to the modern understanding. The major divergence is that Brahmagupta attempted to define division by zero, which is left undefined in modern mathematics. His definition of zero as a number was accurate except that he considered 0/0 equal to 0, whereas modern mathematicians insist that this quantity cannot be defined.
In 628 CE, Brahmagupta gave the first general solution of the quadratic equation:
:$ax^2+bx=c$
This is equivalent to:
:
''Brahmasphutasiddhanta'' has four and a half chapters devoted to pure mathematics while the twelfth chapter, the Ganita, deals with arithmetic progressions and some geometry. The eighteenth chapter of Brahmagupta's work is called the Kuttaka. This is usually associated with the Aryabhata's method for solving the Diophantine equation ''ax'' − ''by'' = ''c''. But here Kuttaka means algebra. Brahmagupta went on to invent a method for solving Diophantine equations of the second degree, such as ''nx''² + 1 = ''y''².
Brahmagupta also gave the formula to find the area of any cyclic quadrilateral given its four sides. Heron's formula is a special case of this formula, when one of the sides equal zero. The relationship between the general Brahmagupta's formula and the Heron's formula is similar to how the law of cosines extends the Pythagorean theorem.

Astronomy

In astronomy, Brahmagupta was the first to use algebra to solve astronomical problems. It was through the ''Brahmasphutasiddhanta'' that the Arabs learned of Indian astronomy. The famous Abbasid caliph Al-Mansur (712–775) founded Baghdad, which is situated on the banks of the Tigris, and made it a center of learning. The caliph invited a scholar of Ujjain by the name of Kankah in 770 A.D. Kankah used the ''Brahmasphutasiddhanta'' to explain the Hindu system of arithmetic astronomy. Al-Fazari translated Brahmugupta's work into Arabic upon the request of the caliph.
Some of the important contributions made by Brahmagupta in astronomy are: methods for calculating the position of heavenly bodies over time (ephemerides), their rising and setting, conjunctions, and the calculation of solar and lunar eclipses. Brahmagupta criticized the Puranic view that the Earth was flat or hollow. Instead, he observed that the Earth and heaven were spherical and that the Earth is moving. In 1030, the Muslim astronomer Abu al-Rayhan al-Biruni, in his ''Ta'rikh al-Hind'', later translated into Latin as ''Indica'', commented on Brahmagupta's work and wrote that critics argued:
According to al-Biruni, Brahmagupta responded to these criticisms with the following argument on gravitation:

Notes