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Yuktibhāṣā (യുക്തിഭാഷ), also known as Gaṇita-yukti-bhāṣā and (English: Compendium of Astronomical Rationale), is a major treatise on mathematics and astronomy, written by the Indian astronomer Jyesthadeva of the Kerala school of mathematics around 1530. The treatise, written in Malayalam, is a consolidation of the discoveries by Madhava of Sangamagrama, Nilakantha Somayaji, Parameshvara, Jyeshtadeva, Achyuta Pisharati, and other astronomer-mathematicians of the Kerala school. It also exists in a Sanskrit version, with unclear author and date, composed as a rough translation of the Malayalam original. The work contains proofs and derivations of the theorems that it presents. Modern historians used to assert, based on the works of Indian mathematics that first became available, that early Indian scholars in astronomy and computation lacked in proofs, but demonstrates otherwise. Some of its important topics include the infinite series expansions of functions; power series, including of π and π/4; trigonometric series of sine, cosine, and arctangent; Taylor series, including second and third order approximations of sine and cosine; radii, diameters and circumferences; and tests of convergence. mainly gives rationale for the results in Nilakantha's Tantra Samgraha. It is considered an early text on the ideas of calculus, predating Newton and Leibniz by two centuries. The treatise was largely unnoticed outside India, as it was written in the local language of Malayalam. In modern times, due to wider international cooperation in mathematics, the wider world has taken notice of the work. For example, both Oxford University and the Royal Society of Great Britain have given attribution to pioneering mathematical theorems of Indian origin that predate their Western counterparts. contains most of the developments of the earlier Kerala school, particularly Madhava and Nilakantha. The text is divided into two parts – the former deals with mathematical analysis and the latter with astronomy.

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Madhava series

In mathematics, a Madhava series is one of the three Taylor series expansions for the sine, cosine, and arctangent functions discovered in 14th or 15th century Kerala by the mathematician and astronomer Madhava of Sangamagrama (c. 1350 – c. 1425) or his followers in the Kerala school of astronomy and mathematics. Using modern notation, these series are: All three series were later independently discovered in 17th century Europe.

Nombre irrationnel

Un nombre irrationnel est un nombre réel qui n'est pas rationnel, c'est-à-dire qu'il ne peut pas s'écrire sous la forme d'une fraction a/b, où a et b sont deux entiers relatifs (avec b non nul). Les nombres irrationnels peuvent être caractérisés de manière équivalente comme étant les nombres réels dont le développement décimal n'est pas périodique ou dont le développement en fraction continue est infini. On distingue, parmi les nombres irrationnels, deux sous-ensembles complémentaires : les nombres algébriques non rationnels et les nombres transcendants.

École du Kerala

thumb|Région du Kerala, Inde L’école du Kerala est une école de mathématiques et d'astronomie fondée par Madhava de Sangamagrama dans la province du Kerala en Inde, et ayant eu entre autres pour membres et Nilakantha Somayaji. Elle prospéra entre le et le , s'achevant avec les travaux de Melpathur Narayana Bhattathiri. Les découvertes mathématiques de l'école anticipent de deux siècles certains des résultats du calcul infinitésimal de Newton et Leibniz (mais non leurs techniques), obtenant par exemple le développement en série entière des fonctions trigonométriques, mais il n'y a pas de preuve que ces découvertes se soient diffusées en dehors du Kerala.