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 exposé 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, Niccolò
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
géometrie (1637), written by the French philosopher and
mathematician René
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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