History of Algebra


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.

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 wrote one of the first Arabic algebras, a systematic exposé of the basic theory of equations, with both examples and proofs. 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.

By the time of Gauss, algebra had entered its modern phase. Attention shifted from solving polynomial equations to the structure of systems of elements that arose as abstractions of systems, such as the complex numbers, that mathematicians had met in their study of polynomial equations. Two examples of such systems are groups (see Group) and quaternions, which share some of the properties of number systems but also depart from them in important ways. Groups began as systems of permutations and combinations of roots of polynomials, but they became one of the chief unifying concepts of 19th-century mathematics. Important contributions to their 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 (1842-99). Quaternions were the discovery of the British mathematician William Rowan Hamilton, who extended the arithmetic of complex numbers (see Complex Number) to quaternions (quadruples of real numbers).

Immediately after Hamilton's discovery, the German mathematician Hermann Grassmann (1809-77) began investigating vectors (see Vector). Despite its abstract character, American physicist J. W. Gibbs recognized in vector algebra a system of great utility for physicists, just as Hamilton had recognized the usefulness of quaternions. The widespread influence of this abstract approach led to the publication in 1854 of George Boole's The Laws of Thought, an algebraic treatment of basic logic. Since that time, modern algebra—also called abstract algebra—has continued to develop. Important new results have been discovered, and the subject has found applications in all branches of mathematics and in many of the sciences, as well.

 

 

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