Childhood & Early Life
Brahmagupta was born in 598 AD into an orthodox Shaivite Hindu family. His father’s name was Jishnugupta. It is generally believed that he was born in Ujjain. Not much is known about his early life.
As a young man he studied astronomy extensively. He was well-read in the five traditional siddhanthas on Indian astronomy, and also studied the work of other ancient astronomers such as Aryabhata I, Latadeva, Pradyumna, Varahamihira, Simha, Srisena, Vijayanandin and Vishnuchandra.
Brahmagupta became an astronomer of the Brahmapaksha school, one of the four major schools of Indian astronomy during his era.
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Later Years
He is believed to have lived and worked in Bhinmal in present day Rajasthan, India, for a few years. The city was a center of learning for mathematics and astronomy, and he flourished as an astronomer in the intellectual atmosphere of the city.
At the age of 30, he composed the theoretical treatise ‘Brāhmasphuṭasiddhānta’ ("Correctly established doctrine of Brahma") in 628 AD. The work is thought to be a revised version of the received siddhanta of the Brahmapaksha school, incorporated with some of his own new material. Primarily a book of astronomy, it also contains several chapters on mathematics.
Brahmagupta is credited to have given the most accurate of the early calculations of the length of the solar year. He initially estimated it to be at 365 days, 6 hours, 5 minutes, and 19 seconds which is remarkably close to the actual value of 365 days, 5 hours, 48 minutes, and about 45 seconds.
He later revised his estimate and proposed a length of 365 days, 6 hours, 12 minutes, and 36 seconds. His work was very significant considering the fact that he had no telescope or scientific equipment to help him arrive at his conclusions. He is believed to have relied primarily on Aryabhata’s findings to arrive at his own conclusions.
In addition to astronomy, his book also contained various chapters on mathematics. Through this book, he laid the foundations of the two major fields of Indian mathematics, pati-ganita (“mathematics of procedures,” or algorithms) and bija-ganita (“mathematics of seeds,” or equations).
The ‘Brāhmasphuṭasiddhānta’ was the first book to mention zero as a number. He further gave rules of using zero with negative and positive numbers. He also described the rules of operations on negative numbers which come quite close to the modern understanding of numbers.
He also introduced new methods for solving quadratic equations and gave equations to solve systems of simultaneous indeterminate equations, in addition to providing two equivalent solutions to the general quadratic equation.
In his seminal book he provided a formula useful for generating Pythagorean triples and also gave a recurrence relation for generating solutions to certain instances of Diophantine equations.
In mathematics, his contribution to geometry was especially significant. His formula for cyclic quadrilaterals--now known as Brahmagupta's formula—provides a way of calculating the area of any cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides.
He gave formulas for the lengths and areas of other geometric figures as well, and the Brahmagupta's theorem named after him states that if a cyclic quadrilateral has perpendicular diagonals, then the perpendicular diagonal to a side from the point of intersection of the diagonals always bisects the opposite side.
One of his later works was the treatise ‘Khaṇḍakhādyaka’ (meaning "edible bite; morsel of food"), written in 665 AD which covered several topics on astronomy including the longitudes of the planets, diurnal rotation, lunar and solar eclipses, risings and settings, the moon's crescent and conjunctions of the planets.