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 some indeterminate equations.
The Alexandrian mathematicians Hero of Alexandria and
Diophantus continued the traditions of Egypt and Babylon, but Diophantus's 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,
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 mathematician, astronomer, and poet
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. In the early 13th century, the great Italian mathematician Leonardo
Fibonacci achieved a close approximation to the solution of the
cubic equation x3 + 2
x2 +
cx =
d. Because Fibonacci had traveled in Islamic lands, 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. Cardano's pupil, Ludovico Ferrari, soon found an exact solution to equations of the fourth degree (see
quartic equation), 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
Evariste 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. 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, showing that every polynomial equation has at least one root in the complex plane (
see Number:
Complex Numbers).
By the time of Gauss, algebra had entered its modern phase. Attention shifted from solving
polynomial equations to studying the structure of abstract mathematical systems whose axioms were based on the behavior of mathematical objects, such as
complex numbers, that mathematicians encountered when studying polynomial equations. Two examples of such systems are
algebraic 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.
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.
Immediately after Hamilton's discovery, the German mathematician
Hermann Grassmann began investigating vectors. 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
George Boole to write
The Laws of Thought (1854), 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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