**History of Algebra**

**The history of algebra began in ancient Egypt and Babylon**, where they were able to solve linear equations (**ax = b**) and quadratic equations (**ax² + bx = c**), as well as indeterminate equations such as (**x²+ y²= z²**), with several unknowns.

The ancient Babylonians solved any second-degree equation using essentially the same methods taught today. They were also able to solve some indeterminate equations.

The Alexandrian mathematicians Heron and Diophant continued the tradition of Egypt and Babylon, although the book Diophant’s Arithmetic is much more advanced and presents many surprising **solutions for difficult indeterminate equations**.

This ancient wisdom on solving equations found, in turn, acceptance in the Islamic world, where it was called **«Science of reduction and equilibrium»**, the Arabic word al-jabru meaning «reduction», is the origin of the **word algebra**.

In the ninth century, the mathematician **Musa al-Khwarizmi** wrote one of the first Arabic Algebra books, a systematic presentation of the Fundamental Theory of Equations, with examples and demonstrations included.

At the end of the 9th century, the Egyptian mathematician Abu Kamil enunciated and demonstrated the fundamental laws and identities of Algebra, and solved problems as complicated as finding the variables «x», «y», «z» that meet: **x + y + z = 10**; **x² + y² = z²**; and **x.z = y²**.

In ancient civilizations algebraic expressions were written using abbreviations only occasionally; however, in the Middle Ages, Arab mathematicians were able to describe any power of the incognito «x», and developed the fundamental algebra of polynomials, though without using modern symbols.

**This algebra included multiplying, dividing, and extracting square roots from polynomials**, as well as knowledge of the **Binomial Theorem**.

The Persian mathematician, poet and astronomer Omar Khayyam showed how to express the roots of cubic equations using the segments obtained by intersection of conical sections, although he was unable to find a formula for the roots. The Latin translation of al-Khwarizmi’s Algebra was published in the 12th century.

At the beginning of the 13th century, the Italian mathematician Leonardo Fibonacci managed to find a close approximation to the solution of the cubic equation: x³ + 2x²+ cx = d. Fibonacci had travelled to Arab countries, so he certainly used the Arabic method of successive approximations.

At the beginning of the 16th century the Italian mathematicians Scipione del Ferro, Tartaglia and Gerolamo Cardano solved the general cubic equation according to the constants that appear in the equation. Ludovico Ferrari, Cardano’s pupil, soon found the exact solution for the fourth-degree equation and, as a consequence, some mathematicians in later centuries tried to find the formula of the roots of the fifth-degree and higher equations.

However, at the beginning of the 19th century the Norwegian mathematician Niels Abely the French Évariste Galois demonstrated the inexistence of such a formula.

An important advance in Algebra was the introduction, in the 16th century, of symbols for incognites and for algebraic operations and powers. Because of this advance, Book III of Geometry (1637), written by the French mathematician and philosopher **René Descartes** resembles a modern Algebra text.

However, Descartes’ most important contribution to Mathematics was the discovery of Geometry which also contains the fundamentals of an equation theory course, including what Descartes himself in the 18th century continued to work on equation theory and in 1799 the German mathematician **Carl Friedrich Gauss** published the demonstration that every polynomial equation has at least one root in the complex plane.

In Gauss’s time, **Algebra had entered its modern stage**. The focus shifted from polynomial equations to the study of the structure of abstract mathematical systems, whose axioms were based on the behavior of mathematical objects, such as complex numbers, that mathematicians had found when studying polynomial equations.

Two examples of such systems are groups and quaterns, which share some of the properties of numerical systems, although they also differ substantially from them.

The groups began as systems of permutations and combinations of the roots of polynomials, but evolved to become one of the most important unifying concepts of mathematics in the nineteenth century.

French **mathematicians** Galois and Augustin Cauchy, British mathematician Arthur Cayley, and Norwegians Niels Abel and Sophus Lie made important contributions to his style. Quaterns were discovered by the Irish mathematician and astronomer William Hamilton, who developed the Arithmetic of complex numbers for quaterns; while complex numbers are of the form a + bi, quaterns are of the form **a + bi + cj + dk**.

After Hamilton’s discovery the German mathematician Hermann Grassmann began to investigate vectors. In spite of his abstract character, the American physicist J. W. Gibbs found in vector algebra a very useful system for physicists, just as Hamilton had done with quaterns.

The broad influence of this abstract approach led George Boole to write Research on the Laws of Thought (1854), an algebraic treatment of basic logic.

Since then, **modern Algebra** – also called **Abstract Algebra** – has continued to evolve; important results have been obtained and applications have been found in all branches of mathematics and in many other sciences.