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India

Geometrical and mathematical problems involving irrational numbers such as square roots were addressed very early during the Vedic period in India. There are references to such calculations in the Samhitas, Brahmanas, and the Shulba Sutras (800 BC or earlier).

It is suggested that the concept of irrationality was implicitly accepted by Indian mathematicians since the 7th century BC, when Manava (c. 750 – 690 BC) believed that the square roots of numbers such as 2 and 61 could not be exactly determined. Historian Carl Benjamin Boyer, however, writes that "such claims are not well substantiated and unlikely to be true".

Later, in their treatises, Indian mathematicians wrote on the arithmetic of surds including addition, subtraction, multiplication, rationalization, as well as separation and extraction of square roots.

Mathematicians like Brahmagupta (in 628 AD) and Bhāskara I (in 629 AD) made contributions in this area as did other mathematicians who followed. In the 12th century Bhāskara II evaluated some of these formulas and critiqued them, identifying their limitations.

During the 14th to 16th centuries, Madhava of Sangamagrama and the Kerala school of astronomy and mathematics discovered the infinite series for several irrational numbers such as π and certain irrational values of trigonometric functions. Jyeṣṭhadeva provided proofs for these infinite series in the Yuktibhāṣā.