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 M
UHAMMAD IBN
M–
US
–
AAL
-K
HW
–
ARIZM
–
I
(ca.
780–850) published his famous piece Hisab al-jabr w’al-
muqa¯ 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 per-
forming 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. None-
theless, 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 F
IBONACCI
(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 G
IROLAMO
C
ARDANO
(1501–76) published Ars
magna (The great art), which included solutions to the cubic
and
QUARTIC EQUATION
s, as well as other mathematical dis-
coveries. 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 F
RANÇOIS
V
IÈTE
(1540–1603) introduced an efficient system for denoting
powers of variables and was the first to use letters as coef-
ficients before variables, as in “ax2+ bx + c,” for instance.
(Viète also introduced the signs “+” and “–,” although he
never used a sign for equality.) R
ENÈ
D
ESCARTES
(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 com-
pletely 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 C
ARL
F
RIEDRICH
G
AUSS
(1777–1855) proved the
FUNDAMENTAL THEOREM OF ALGEBRA
in
1797, which states that every
POLYNOMIAL
equation of degree
nhas at least one and at most n(possibly complex) roots.
His work, however, does not provide actual methods for
finding these roots.
Renaissance scholars S
CIPIONE DEL
F
ERRO
(1465–1526)
and N
ICCOLÒ
T
ARTAGLIA
(ca. 1500–57) both knew how to solve
cubic equations, and in his 1545 treatise Ars magna, Car-
dano published the solution to the quartic equation discov-
ered by his assistant L
UDOVICO
F
ERRARI
(1522–65). For the
centuries that followed, mathematicians attempted to find a
general arithmetic method for solving all quintic (fifth-
degree) equations. L
EONHARD
E
ULER
(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 sug-
gested the same, and just a few years later Norwegian
mathematician N
IELS
H
ENRIK
A
BEL
(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 alge-
braically. (Equation of the form x5– a= 0, for instance, have
solutions x= 5√a.
–
) In 1831 French mathematician É
VARISTE
G
ALOIS
(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 vari-
ables to represent quantities other than real numbers. For
example, English mathematician G
EORGE
B
OOLE
(1815–64)
invented an algebra symbolic logic in which variables rep-
resented sets, and Irish scholar S
IR
W
ILLIAM
R
OWAN
H
AMIL
-
TON
(1805–65) invented algebraic systems in which
variables represented
VECTOR
s or
QUATERNION
s.
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×bmight not necessarily give the same result as
b×a. This motivated mathematicians to develop abstract
AXIOM
s to explain the workings of different algebraic sys-
tems. Thus the topic of
ABSTRACT ALGEBRA
was born. One
outstanding contributor in this field was German mathemati-
cian A
MALIE
N
OETHER
(1883–1935), who made important dis-
coveries about the nature of noncommutative algebras.
See also
ASSOCIATIVE
; B
ABYLONIAN MATHEMATICS
;
CANCEL
-
LATION
;
COMMUTATIVE PROPERTY
; E
GYPTIAN MATHEMATICS
;
FIELD
;
G
REEK MATHEMATICS
; I
NDIAN MATHEMATICS
;
LINEAR ALGEBRA
;
RING
.
History of Equations and Algebra
(continued)