<P> Earlier traditions discussed above had a direct influence on the Persian mathematician Muḥammad ibn Mūsā al - Khwārizmī (c. 780--850). He later wrote The Compendious Book on Calculation by Completion and Balancing, which established algebra as a mathematical discipline that is independent of geometry and arithmetic . </P> <P> The Hellenistic mathematicians Hero of Alexandria and Diophantus as well as Indian mathematicians such as Brahmagupta continued the traditions of Egypt and Babylon, though Diophantus' Arithmetica and Brahmagupta's Brāhmasphuṭasiddhānta are on a higher level . For example, the first complete arithmetic solution (including zero and negative solutions) to quadratic equations was described by Brahmagupta in his book Brahmasphutasiddhanta . Later, Persian and Arabic mathematicians developed algebraic methods to a much higher degree of sophistication . Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Al - Khwarizmi's contribution was fundamental . He solved linear and quadratic equations without algebraic symbolism, negative numbers or zero, thus he had to distinguish several types of equations . </P> <P> In the context where algebra is identified with the theory of equations, the Greek mathematician Diophantus has traditionally been known as the "father of algebra" but in more recent times there is much debate over whether al - Khwarizmi, who founded the discipline of al - jabr, deserves that title instead . Those who support Diophantus point to the fact that the algebra found in Al - Jabr is slightly more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al - Jabr is fully rhetorical . Those who support Al - Khwarizmi point to the fact that he introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation) which the term al - jabr originally referred to, and that he gave an exhaustive explanation of solving quadratic equations, supported by geometric proofs, while treating algebra as an independent discipline in its own right . His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study". He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems". </P> <P> Another Persian mathematician Omar Khayyam is credited with identifying the foundations of algebraic geometry and found the general geometric solution of the cubic equation . His book Treatise on Demonstrations of Problems of Algebra (1070), which laid down the principles of algebra, is part of the body of Persian mathematics that was eventually transmitted to Europe . Yet another Persian mathematician, Sharaf al - Dīn al - Tūsī, found algebraic and numerical solutions to various cases of cubic equations . He also developed the concept of a function . The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al - Karaji, and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher - order polynomial equations using numerical methods . In the 13th century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European algebra . Abū al - Ḥasan ibn ʿAlī al - Qalaṣādī (1412--1486) took "the first steps toward the introduction of algebraic symbolism". He also computed ∑ n, ∑ n and used the method of successive approximation to determine square roots . As the Islamic world was declining, the European world was ascending . And it is here that algebra was further developed . </P>

Who is known as the father of algebra
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