Brahmagupta Indian Mathematician and Astronomer

Brahmagupta (born c. 598, died after 665) was an Indian mathematician and astronomer. He revolutionized both math and astronomy in the 7th century. He was the first to give rules to compute with zero. He is the author of two early works on mathematics and astronomy: the Brahmasphutasiddhanta , a theoretical treatise, and the Khaṇḍakhadyaka.

Brahmagupta was born in 598 CE according to his own statement. He lived in Bhillamala ( Bhinmal ) during the reign of the Chapa dynasty ruler Vyagrahamukha. He was the son of Jishnugupta. He was a Shaivite by religion. Even though scholars assume that he was born in Bhillamala, there is no conclusive evidence for it. At an age of 30, he composed Brāhmasphuṭasiddhānta (the improved treatise of Brahma) which is believed to be a revised version of the received siddhanta of the Brahmapaksha school.

Later, Brahmagupta moved to Ujjain, which was an ancient city in India. An interesting fact about Ujjain was major centre for astronomy. Ujjain in its day was a hub for the study of mathematics and astronomy; basically, a mediation point of the importance of time and its understanding. He lived beyond 665 CE. He is believed to have died in Ujjain.

Brahmagupta was a the great mathematician all time.

Brahmagupta was the first to use zero as a number. He gave rules to compute with zero. Brahmagupta used negative numbers and zero for computing. The modern rule that two negative numbers multiplied together equals a positive number first appears in Brahmasputa siddhanta.

Algebra

Brahmagupta gave the solution of the general linear equation in Brahmasphutasiddhanta,

“The difference between rupas, when inverted and divided by the difference 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 .

He further gave two equivalent solutions to the general quadratic equation:

the equation ax2 + bx = c equivalent to

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] the middle [number].

Whatever is the square-root of the rupas multiplied by the square [and] increased by the square of half the unknown, diminish that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown.

Arithmetic

In Brahmasphutasiddhanta, Multiplication was named Gomutrika. He explained how to find the cube and cube-root of an integer and gave rules facilitating the computation of squares and square roots.

He also gave rules for dealing with five types of combinations of fractions –  a/c + b/c; a/c × b/d; a/1 + b/d; a/c + b/d × a/c = a(d + b)/cd; and a/cb/d × a/c = a(db)/cd .

He gave 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.

Zero

The most notable innovations that Brahmagupta is remembered for is his look and pursuit of the number zero. Until this time, the zero had not been thought of as a number. He formulated equations that allowed zero to be used in positives and negatives. Although he did not use those specific words, Brahmagupta did follow with the importance in the need of the zero placement. Without the use of zero and its value defined, according to him, arithmetic really had nowhere to go.

Brahmagupta describes operations on negative numbers. He first describes addition and subtraction,

The sum of two positives is positives, and sum of two negatives negative; of 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 positive, [and that] of two zeros zero.

A negative minus zero is negative, a positive [minus zero] positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added.

Brahmagupta’s Look at Zero listed:

  • Zero minus zero is a zero.
  • The product of zero multiplied by zero is zero.
  • A debt minus zero is a debt.
  • A fortune minus zero is a fortune.
  • A debt subtracted from zero is a fortune.
  • fortune subtracted from zero is a debt.
  • The product of zero multiplied by a debt or fortune is zero.

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.

Brahmagupta’s theorem

Brahmagupta’s Formula

Brahmagupta dedicated a substantial portion of his work to geometry and trigonometry. He established √10 (3.162277) as a good practical approximation for π (3.141593), and gave a formula, now known as Brahmagupta’s Formula, for the area of a cyclic quadrilateral

Brahmagupta’s most famous result in geometry 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.

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 the sides diminished by [each] side of the quadrilateral.

Brahmagupta’s theorem

The square-root of the sum of the two products of the sides and opposite sides of a non-unequal quadrilateral is the diagonal. The square of the diagonal is diminished by the square of half the sum of the base and the top; the square-root is the perpendicular [altitudes].

 

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    √(t − p)(t − q)(t − r)(t − s).

 

Astronomy

Brahmagupta loved the studies of the heavens and became an astronomer of one of four major Indian astronomy schools; the Brahmapaksha. His studies followed some of the notables in Indian astronomy such as Varahmihara, Aryabhata and Simha.

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.

He explained that since the Moon is closer to the Earth than the Sun, the degree of the illuminated part of the Moon depends on the relative positions of the Sun and the Moon, and this can be computed from the size of the angle between the two bodies.[

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