courtesy : Algebra
History
Main articles: History of algebra, Abstract algebra § History, and Timeline of algebra
The use of the word “algebra” for denoting a part of mathematics dates probably from the 16th century. The word is derived from the Arabic word al-jabr that appears in the title of the treatise Al-Kitab al-muhtasar fi hisab al-gabr wa-l-muqabala (The Compendious Book on Calculation by Completion and Balancing), written circa 820 by Al-Kwarizmi.
Al-jabr referred to a method for transforming equations by subtracting like terms from both sides, or passing one term from one side to the other, after changing its sign.
Therefore, algebra referred originally to the manipulation of equations, and, by extension, to the theory of equations. This is still what historians of mathematics generally mean by algebra.
In mathematics, the meaning of algebra has evolved after the introduction by François Viète of symbols (variables) for denoting unknown or incompletely specified numbers, and the resulting use of the mathematical notation for equations and formulas. So, algebra became essentially the study of the action of operations on expressions involving variables. This includes but is not limited to the theory of equations.
At the beginning of the 20th century, algebra evolved further by considering operations that act not only on numbers but also on elements of so-called mathematical structures such as groups, fields and vector spaces. This new algebra was called modern algebra by van der Waerden in his eponymous treatise, whose name has been changed to Algebra in later editions.
Early history
A page from Al-Khwārizmī’s al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala
The roots of algebra can be traced to the ancient Babylonians, who developed an advanced arithmetical system with which they were able to do calculations in an algorithmic fashion. The Babylonians developed formulas to calculate solutions for problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations. By contrast, most Egyptians of this era, as well as Greek and Chinese mathematics in the 1st millennium BC, usually solved such equations by geometric methods, such as those described in the Rhind Mathematical Papyrus, Euclid’s Elements, and The Nine Chapters on the Mathematical Art. The geometric work of the Greeks, typified in the Elements, provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations, although this would not be realized until mathematics developed in medieval Islam.
By the time of Plato, Greek mathematics had undergone a drastic change. The Greeks created a geometric algebra where terms were represented by sides of geometric objects, usually lines, that had letters associated with them. Diophantus (3rd century AD) was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, and have led, in number theory, to the modern notion of Diophantine equation.
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.
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 written in words instead of symbols, including zero and negative solutions, to quadratic equations was described by Brahmagupta in his book Brahmasphutasiddhanta, published in 628 AD. Later, Persian and Arab 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.
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” and in the context where it is identified with rules for manipulating and solving equations, Persian mathematician al-Khwarizmi is regarded as “the father of algebra”. It is open to debate whether Diophantus or al-Khwarizmi is more entitled to be known, in the general sense, as “the father of algebra”. 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”.
