Indian astronomer and mathematician.

**Born:** c. 598 AD in Billamala

**Died:** c. 670 AD in Ujjain

Brahmagupta is an Indian mathematician and astronomer who made pivotal contributions to arithmetic, algebra, geometry, and the study of quadratic indeterminate equations. His works were of great importance to later Indian mathematicians who would draw from his discoveries to perfect the generalized solution of Pell's equation. Brahmagupta was also the first recorded mathematician to have demystified the properties of the number zero to its modern form.

## **Discoveries**

**Astronomical Theory**

Brahmagupta opposed the ancient Vedic astronomers who argued that the Sun is closer to the Earth than the moon.

**Zero**

In his magnum opus *Brahmasphutasiddhanta*, Brahmagupta was the first known mathematician to document on the arithmetic properties of zero as a number with its own merit. The concept of zero as a number may have been known in India prior to Brahmagupta. Radiocarbon dating of the *Bakhshali Manuscript* (which includes the symbol for zero) has revealed that parts of the document was written in the 4th century AD, the 7th century AD, and the 10th century AD.

**Approximation of Pi**

Brahmagupta used two approximations of pi:

- $ \pi = 3 $
- $ \pi = \sqrt{10} $

**Pell's Equation**

Brahmagupta solved certain quadratic indeterminate equations of the form $ Nx^2 + 1 = y^2 $, which is called Pell's equation in the West. Pell's equation was intensively studied in ancient India after Brahmagupta, and his contributions paved the road to the general solution by Bhaskara II in the 12th century AD.

**Cyclic Quadrilaterals**

Brahmagupta was famous for his studies on cyclic quadrilaterals. He discovered the formula for computing the area of a cyclic quadrilateral with sides $ a, b, c, d $

- $ A = \sqrt{s(s-a)(s-b)(s-c)(s-d)} $

where $ s = \frac{a+b+c+d}{2} $ is the semi-perimeter.

He also discovered the property of orthodiagonal cyclic quadrilaterals, which is called the Brahmagupta theorem.