10 algebra
quadratic equations and outlined general methods for solving
systems of equations containing several variables. (He also
had a clear understanding of negative numbers and was
comfortable working with zero as a valid numerical quantity.)
The scholar Bh – askara (ca. 1114–85) used letters to represent
unknown quantities and, in working with quadratic equations,
suggested that all positive numbers have two square roots
and that negative numbers have no (meaningful) roots.
A significant step toward the development of modern
algebra occurred in Baghdad, Iraq, in the year 825 when the
Arab mathematician MUHAMMAD IBN M –
US
– A AL-KHW – ARIZM – I (ca.
780–850) published his famous piece Hisab al-jabr w’almuqa ¯ bala (Calculation by restoration and reduction). This
work represents the first clear and complete exposition on
the art of solving linear equations by a new practice of performing the same operation on both sides of an equation. For
example, the expression x – 3 = 7 can be “restored” to x = 10
by adding three to both sides of the expression, and the
equation 5x = 10 can be “reduced” to x = 2 by dividing both
sides of the equation by five. Al-Khw –
arizm – ı also showed how
to solve quadratic equations via similar techniques. His
descriptions, however, used no symbols, and like the ancient
Greeks, al-Khw –
arizm – ı wrote everything out in words. Nonetheless, al-Khw –
arizm – ı’s treatise was enormously influential,
and his new approach to solving equations paved the way for
modern algebraic thinking. In fact, it is from the word al-jabr
in the title of his book that our word algebra is derived.
Al-Khw – arizm – ı’s work was translated into Latin by the
Italian mathematician FIBONACCI (ca. 1175–1250), and his
efficient methods for solving equations quickly spread
across Europe during the 13th century. Fibonacci translated
the word shai used by al-Khw – arizm – ı for “the thing
unknown” into the Latin term res. He also used the Italian
word cosa for “thing,” and the art of algebra became
known in Europe as “the cossic art.”
In 1545 GIROLAMO CARDANO (1501–76) published Ars
magna (The great art), which included solutions to the cubic
and QUARTIC EQUATIONs, as well as other mathematical discoveries. By the end of the 17th century, mathematicians
were comfortable performing the same sort of symbolic
manipulations we practice today and were willing to accept
negative numbers and irrational quantities as solutions to
equations. The French mathematician FRANÇOIS VIÈTE
(1540–1603) introduced an efficient system for denoting
powers of variables and was the first to use letters as coefficients before variables, as in “ax
2 + bx + c,” for instance.
(Viète also introduced the signs “+” and “–,” although he
never used a sign for equality.) RENÈ DESCARTES (1596–1650)
introduced the convention of denoting unknown quantities
by the last letters of the alphabet, x, y, and z, and known
quantities by the first, a, b, c. (This convention is now completely ingrained; when we see, for example, an equation of
the form ax + b = 0, we assume, without question, that it is
for “x” we must solve.)
The German mathematician CARL FRIEDRICH GAUSS
(1777–1855) proved the FUNDAMENTAL THEOREM OF ALGEBRA in
1797, which states that every POLYNOMIAL equation of degree
n has at least one and at most n (possibly complex) roots.
His work, however, does not provide actual methods for
finding these roots.
Renaissance scholars SCIPIONE DEL FERRO (1465–1526)
and NICCOLÒ TARTAGLIA (ca. 1500–57) both knew how to solve
cubic equations, and in his 1545 treatise Ars magna, Cardano published the solution to the quartic equation discovered by his assistant LUDOVICO FERRARI (1522–65). For the
centuries that followed, mathematicians attempted to find a
general arithmetic method for solving all quintic (fifthdegree) equations. LEONHARD EULER (1707–83) suspected
that the task might be impossible. Between the years 1803
and 1813, Italian mathematician Paolo Ruffini (1765–1822)
published a number of algebraic results that strongly suggested the same, and just a few years later Norwegian
mathematician NIELS HENRIK ABEL (1802–29) proved that,
indeed, there is no general formula that solves all quintic
equations in a finite number of arithmetic operations. Of
course, some degree-five equations can be solved algebraically. (Equation of the form x
5 – a = 0, for instance, have
solutions x =
5
√a.
– ) In 1831 French mathematician ÉVARISTE
GALOIS (1811–32) completely classified those equations that
can be so solved, developing work that gave rise to a whole
new branch of mathematics today called GROUP THEORY.
In the 19th century mathematicians began using variables to represent quantities other than real numbers. For
example, English mathematician GEORGE BOOLE (1815–64)
invented an algebra symbolic logic in which variables represented sets, and Irish scholar SIR WILLIAM ROWAN HAMILTON (1805–65) invented algebraic systems in which
variables represented VECTORs or QUATERNIONs.
With these new systems, important characteristics of
algebra changed. Hamilton, for instance, discovered that
multiplication was no longer commutative in his systems: a
product a × b might not necessarily give the same result as
b × a. This motivated mathematicians to develop abstract
AXIOMs to explain the workings of different algebraic systems. Thus the topic of ABSTRACT ALGEBRA was born. One
outstanding contributor in this field was German mathematician AMALIE NOETHER (1883–1935), who made important discoveries about the nature of noncommutative algebras.
See also ASSOCIATIVE; BABYLONIAN MATHEMATICS; CANCELLATION; COMMUTATIVE PROPERTY; EGYPTIAN MATHEMATICS; FIELD;
GREEK MATHEMATICS; INDIAN MATHEMATICS; LINEAR ALGEBRA; RING.
History of Equations and Algebra
(continued)
quadratic equations and outlined general methods for solving
systems of equations containing several variables. (He also
had a clear understanding of negative numbers and was
comfortable working with zero as a valid numerical quantity.)
The scholar Bh – askara (ca. 1114–85) used letters to represent
unknown quantities and, in working with quadratic equations,
suggested that all positive numbers have two square roots
and that negative numbers have no (meaningful) roots.
A significant step toward the development of modern
algebra occurred in Baghdad, Iraq, in the year 825 when the
Arab mathematician MUHAMMAD IBN M –
US
– A AL-KHW – ARIZM – I (ca.
780–850) published his famous piece Hisab al-jabr w’almuqa ¯ bala (Calculation by restoration and reduction). This
work represents the first clear and complete exposition on
the art of solving linear equations by a new practice of performing the same operation on both sides of an equation. For
example, the expression x – 3 = 7 can be “restored” to x = 10
by adding three to both sides of the expression, and the
equation 5x = 10 can be “reduced” to x = 2 by dividing both
sides of the equation by five. Al-Khw –
arizm – ı also showed how
to solve quadratic equations via similar techniques. His
descriptions, however, used no symbols, and like the ancient
Greeks, al-Khw –
arizm – ı wrote everything out in words. Nonetheless, al-Khw –
arizm – ı’s treatise was enormously influential,
and his new approach to solving equations paved the way for
modern algebraic thinking. In fact, it is from the word al-jabr
in the title of his book that our word algebra is derived.
Al-Khw – arizm – ı’s work was translated into Latin by the
Italian mathematician FIBONACCI (ca. 1175–1250), and his
efficient methods for solving equations quickly spread
across Europe during the 13th century. Fibonacci translated
the word shai used by al-Khw – arizm – ı for “the thing
unknown” into the Latin term res. He also used the Italian
word cosa for “thing,” and the art of algebra became
known in Europe as “the cossic art.”
In 1545 GIROLAMO CARDANO (1501–76) published Ars
magna (The great art), which included solutions to the cubic
and QUARTIC EQUATIONs, as well as other mathematical discoveries. By the end of the 17th century, mathematicians
were comfortable performing the same sort of symbolic
manipulations we practice today and were willing to accept
negative numbers and irrational quantities as solutions to
equations. The French mathematician FRANÇOIS VIÈTE
(1540–1603) introduced an efficient system for denoting
powers of variables and was the first to use letters as coefficients before variables, as in “ax
2 + bx + c,” for instance.
(Viète also introduced the signs “+” and “–,” although he
never used a sign for equality.) RENÈ DESCARTES (1596–1650)
introduced the convention of denoting unknown quantities
by the last letters of the alphabet, x, y, and z, and known
quantities by the first, a, b, c. (This convention is now completely ingrained; when we see, for example, an equation of
the form ax + b = 0, we assume, without question, that it is
for “x” we must solve.)
The German mathematician CARL FRIEDRICH GAUSS
(1777–1855) proved the FUNDAMENTAL THEOREM OF ALGEBRA in
1797, which states that every POLYNOMIAL equation of degree
n has at least one and at most n (possibly complex) roots.
His work, however, does not provide actual methods for
finding these roots.
Renaissance scholars SCIPIONE DEL FERRO (1465–1526)
and NICCOLÒ TARTAGLIA (ca. 1500–57) both knew how to solve
cubic equations, and in his 1545 treatise Ars magna, Cardano published the solution to the quartic equation discovered by his assistant LUDOVICO FERRARI (1522–65). For the
centuries that followed, mathematicians attempted to find a
general arithmetic method for solving all quintic (fifthdegree) equations. LEONHARD EULER (1707–83) suspected
that the task might be impossible. Between the years 1803
and 1813, Italian mathematician Paolo Ruffini (1765–1822)
published a number of algebraic results that strongly suggested the same, and just a few years later Norwegian
mathematician NIELS HENRIK ABEL (1802–29) proved that,
indeed, there is no general formula that solves all quintic
equations in a finite number of arithmetic operations. Of
course, some degree-five equations can be solved algebraically. (Equation of the form x
5 – a = 0, for instance, have
solutions x =
5
√a.
– ) In 1831 French mathematician ÉVARISTE
GALOIS (1811–32) completely classified those equations that
can be so solved, developing work that gave rise to a whole
new branch of mathematics today called GROUP THEORY.
In the 19th century mathematicians began using variables to represent quantities other than real numbers. For
example, English mathematician GEORGE BOOLE (1815–64)
invented an algebra symbolic logic in which variables represented sets, and Irish scholar SIR WILLIAM ROWAN HAMILTON (1805–65) invented algebraic systems in which
variables represented VECTORs or QUATERNIONs.
With these new systems, important characteristics of
algebra changed. Hamilton, for instance, discovered that
multiplication was no longer commutative in his systems: a
product a × b might not necessarily give the same result as
b × a. This motivated mathematicians to develop abstract
AXIOMs to explain the workings of different algebraic systems. Thus the topic of ABSTRACT ALGEBRA was born. One
outstanding contributor in this field was German mathematician AMALIE NOETHER (1883–1935), who made important discoveries about the nature of noncommutative algebras.
See also ASSOCIATIVE; BABYLONIAN MATHEMATICS; CANCELLATION; COMMUTATIVE PROPERTY; EGYPTIAN MATHEMATICS; FIELD;
GREEK MATHEMATICS; INDIAN MATHEMATICS; LINEAR ALGEBRA; RING.
History of Equations and Algebra
(continued)