Algebra- that branch of mathematics dealing with properties of numbers and quantities by means of letters and other general symbols.
[al-jabr al the, jabr reunion of broken parts (Arabian)]

The history of algebra began in ancient Egypt and Babylon, 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. They also could solve indeterminate equations.

About 2000 years ago the Babylonians used the formula ab = (a+b)²/4-(a-b)²/4 to make multiplication easier.
This shows that a table of squares is all that is necessary to multiply numbers. You take the difference of two numbers that are looked up in the table.

The Alexandrian mathematicians Hero 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, and algebra as a science is an Arabic contribution.)

In the 9th century, al-Khwarizmi (born about 780 in Baghdad, died about 850) wrote one of the first Arabic algebras, a systematic exposé of the basic theory of equations, with both examples and proofs, including "completing the square", though he used only words and no symbols in his written work. By the end of the 9th century the Egyptian Abu Kamil (850-930) 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, x² + 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 work out the basic algebra of polynomials (without yet using modern symbolism). This included the ability to multiply, divide, and find square roots of polynomials as well as a knowledge of the binomial theorem.

The Persian Omar Khayyam showed how to express roots of cubic equations by 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, and in the early 13th century appeared the writings of the great Italian mathematician Leonardo Fibonacci (1170-1230), among whose achievements was a close approximation to the solution of the cubic equation x³ + 2x² + cx = d. Because Fibonacci had travelled in Islamic lands, he probably used an Arabic method of successive approximations.

Early in the 16th century, the Italian mathematicians Scipione del Ferro (1465-1526), Niccolò Tartaglia (1500-57), and Gerolamo Cardano (1501-76) solved the general cubic equation in terms of the constants appearing in the equation. Cardano's pupil, Ludovico Ferrari (1522-65), 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 Niels Abel and the French Évariste 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.

Thus Book III of the French philosopher and mathematician René Descartes' La géometrie (1637) looks much like a modern algebra text. Descartes is most significant in mathematics, however, for his discovery of analytic geometry, which reduces the solution of geometric problems to the solution of algebraic ones. His geometry text also contained the essentials of a course on the theory of equations, including his so-called rule of signs for counting the number of what Descartes called the “true” (positive) and “false” (negative) roots of an equation. Work continued through the 18th century on the theory of equations, but not until 1799 was the proof published, by the German mathematician Carl Friedrich Gauss, that every polynomial equation has at least one root.

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