Brahmagupta

Brahmagupta (ब्रह्मगुप्त) (Brahmagupta pronounced^ⓘ) (589–668) was an Indian mathematician and astronomer. Brahmagupta was born in 598 A.D. in Bhinmal city in the states 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.

Mathematics

Brahmagupta has been considered the father of arithmetic, algebra, and numerical analysis. The modern arithmetic used today spread from India to Arabia and then to Europe. Initially, it was known as Al Hind in Arabic and De Numero Indorum in Latin. De Numero Indorum means "method of the Indians" and has become our arithmetic and algebra replacing the earlier Roman numerals and abacus based methods. Addition, subtraction, division and other fundamental operations using Hindu Arabic numerals first appears in Brahmasputha Siddhanta which was translated to Arabic as "Sindhind". He also is known to have passed on knowledge to his students "orally" through expressions. He also tended to miss out 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. But orally transmitted presents several challenges in verifying the source of knowledge, as such no proofs are given as to how Brahmagupta discovered these formulae.[1] But 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. 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 considered0/0 was equal to 0. 0/0 still baffles the mathematicians and today lame statement is made to suggest that 0/0 can not be defined.

In 628 CE, Brahmagupta gave the first general solution of the quadratic equation:

$ax^{2}+bx=c$

To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value. (Brahmasphutasiddhanta, Colebrook translation, 1817, page 346)

This is equivalent to:

$x={\frac {{\sqrt {4ac+b^{2}}}-b}{2a}}$

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 nx² + 1 = y².

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

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 like a bowl. Instead, he observed that the Earth and heaven were spherical. However he wrongly believed that the Earth did not move.

See also

• Brahmagupta's formula
• Brahmagupta's identity
• Brahmagupta's theorem
• Brahmagupta interpolation formula
• Brahmagupta matrix
• Brahmagupta's problem
• Indian mathematics
• List of Indian mathematicians

External links

• O'Connor, John J.; Robertson, Edmund F., "Brahmagupta", MacTutor History of Mathematics Archive, University of St Andrews
• Brahmagupta's theorem
• Brahmagupta's theorem II
• Brahmagupta's formula