Algebra is a branch of mathematics in which symbols represent relationships. Classical algebra grew out of methods of solving equations, and it represents numbers with symbols that combine according to the basic arithmetical operations of addition, subtraction, multiplication, division, and the extraction of roots.

Modern algebra has evolved from classical algebra by increasing its attention to the structures within mathematics. Mathematicians consider modern algebra to be a set of objects with rules for connecting or relating them. As such, in its most general form, algebra may commonly be described as the language of mathematics.

The history of algebra began in ancient Egypt and Babylonia, where people learned to solve linear (ax = b) and quadratic (ax2 + bx = c) equations, as well as indeterminate equations such as x2 + y2 = z2, whereby several unknowns are involved. The ancient Babylonians solved arbitrary quadratic equations by essentially the same procedures taught today.

The Alexandrian mathematicians Hero of Alexandria and Diophantus continued the traditions of Egypt and Babylon, but Diophantus’ book Arithmetica is on a much higher level and gives many surprising solutions to difficult indeterminate equations.

This ancient knowledge of solutions of equations in turn found a home early in the Islamic world, where it was known as the “science of restoration and balancing”. The Arabic word for restoration, al-jabru, is the root of the word “algebra”.

In the 9th century, the Arab mathematician al-Khwarizmi wrote one of the first Arabic algebras, a systematic expos? of the basic theory of equations, with both examples and proofs. By the end of the 9th century, the Egyptian mathematician Abu Kamil had stated and proved the basic laws and identities of algebra and solved such complicated problems as finding x, y, and z such that x + y + z = 10, x 2 + y2 = z2, and xz = y2.

Ancient civilizations wrote out algebraic expressions using only occasional abbreviations, but by medieval times Islamic mathematicians were able to talk about arbitrarily high powers of the unknown x, and, without yet using modern symbolism, work out the basic algebra of polynomials. This included the ability to multiply, divide, and find square roots of polynomials as well as a knowledge of the binomial theorem.

The Persian mathematician, astronomer, and poet Omar Khayyam showed how to express roots of cubic equations by means of line segments obtained by intersecting conic sections, but he could not find a formula for the roots. A Latin translation of Al-Khwarizmi’s Algebra appeared in the 12th century. In the early 13th century, the Italian mathematician Leonardo Fibonacci achieved a close approximation to the solution of the cubic equation x3 + 2x2 + cx = d. Because Fibonacci had travelled in Islamic countries, he probably used an Arabic method of successive approximations.

Early in the 16th century, the Italian mathematicians Scipione del Ferro, Niccol? Tartaglia, and Gerolamo Cardano solved the general cubic equation in terms of the constants appearing in the equation. Tartaglia and Cardano’s pupil, Ludovico Ferrari, soon found an exact solution to equations of the fourth degree, and as a result, mathematicians for the next several centuries tried to find a formula for the roots of equations of degree five, or higher. Early in the 19th century, however, the Norwegian mathematician Niels Abel and the French mathematician ?variste Galois proved that no such formula exists.

An important development in algebra in the 16th century was the introduction of symbols for the unknown and for algebraic powers and operations. As a result of this development, Book III of La G?ometrie (1637), written by the French philosopher and mathematician Ren? Descartes, looks much like a modern algebra text. Descartes’s most significant contribution to mathematics, however, was his discovery of analytic geometry, which reduces the solution of geometric problems to the solution of algebraic ones.

In 1799 the German mathematician Carl Friedrich Gauss published the proof showing that every polynomial equation has at least one root in the complex plane. By the time of Gauss, algebra had entered its modern phase, when the attention shifted from solving polynomial equations to studying the structure of abstract mathematical systems whose axioms were based on the behaviour of mathematical objects, such as complex numbers.

Important contributions to algebra study were made by the French mathematicians Galois and Augustin Cauchy, the British mathematician Arthur Cayley, and the Norwegian mathematicians Niels Abel and Sophus Lie. Quaternions were discovered by British mathematician and astronomer William Rowan Hamilton, who extended the arithmetic of complex numbers to quaternions; while complex numbers are of the form a + bi, quaternions are of the form a + bi + cj + dk.

Resources:
Algebra online
Encyclopedia of algebra