Furthermore, we say the ring is commutative if an
eighth axiom holds:
8. Commutative Law for Multiplication: For all elements a and b of R, we have: a*b = b*a.
For example, the set of integers under ordinary addition
and multiplication is a commutative ring. Thus any
result that is known to follow abstractly from the eight
principles outlined above translates to a result about
numbers. The set of all functions f from the set of real
numbers to itself also satisfies the definition of a commutative ring, and so these results also translate to
interesting facts in function theory. As a simple example, it is straightforward to prove that the zero element
in a ring is unique. (If 0 and 0′ are both zeros, then, by
axiom three, 0 = 0 + 0′ = 0′.) Consequently, there is also
only one function that can behave as the zero function.
One can impose further conditions on a system.
For example, a commutative ring is called an integral
domain if a ninth axiom holds:
9. No Divisors of Zero: It is never the case that two
nonzero elements a and b of R give a*b = 0.
The set of integers is an integral domain, but the
set of functions is not. For example, if f is the function
that gives the value zero for negative inputs, and the
value 1 otherwise, and g is the function that gives the
value zero for positive inputs, and the value 1 otherwise, then neither f nor g is the zero function, but their
product f*g is. Any system in MODULAR ARITHMETIC
forms a ring but not necessarily an integral domain.
For example, 2 × 3 = 0 modulo 6.
A commutative ring is called a field if, further, a
10th axiom is satisfied:
10. Existence of Multiplicative Inverses: For each
nonzero element a of R, there is an element b such
that a*b = b*a = 1.
The set of integers is not a field. (The number 2,
for example, has no multiplicative inverse, since 1/2 is
not an integer.) The set of all real numbers under
addition and multiplication, however, is a field, as is
the set of all the rational numbers and the set of all
complex numbers.
It is usually assumed that the elements 0 and 1 in a
commutative ring or a field are different. If these two
elements are the same, that is, if 0 = 1, then one can
prove that the ring R contains only this element: R =
{0}. This ring is called the trivial ring.
See also ABSTRACT ALGEBRA; ASSOCIATIVE; COMMUTATIVE PROPERTY; DISTRIBUTIVE PROPERTY; GROUP;
GROUP THEORY; QUATERNIONS; ZERO.
Rolle, Michel (1652–1719) French Calculus Born on
April 21, 1652, in Ambert, France, Michel Rolle is best
remembered for the theorem in CALCULUS that bears his
name. Although attracting little attention at the time of
its publication, Rolle’s theorem is today considered one
of the fundamental principles of the subject.
Rolle began his career as a scribe and as an assistant to an attorney. He had little formal education but
pursued a personal interest in mathematics all his life.
He moved to Paris in 1675 and soon developed a reputation as an expert in arithmetic work. In 1682 he
achieved national fame for solving a recreational mathematics problem publicly posed by French mathematician Jacques Ozanam. In honor of this achievement,
Rolle was awarded a pension by France’s controller of
general finance and admission to the Académie Royal
des Sciences in 1685.
In 1690 he published the work Traité d’algèbre
(Treatise on algebra), a text on the theory of equations,
in which, among other things, he invented and used the
notation
n
√
–
x for the nth root of x. Rolle also studied
GEOMETRY, ALGEBRA, and DIOPHANTINE EQUATIONs.
His famous theorem was published one year later in an
obscure and little noticed text Démonstration d’une
méthode pour resoudre les égalitez de tous les degrez
(Proof of a method for solving equations of all degrees).
It is ironic that Rolle is today considered a principal figure in the development of calculus. Having studied the emerging subject, Rolle is said to have found
the theory unconvincing. He even went so far as to say
that calculus is nothing more than a “collection of
ingenious fallacies.” He died in Paris, France, on
November 8, 1719.
See also ROLLE’S THEOREM.
Rolle’s theorem In 1691 French mathematician
MICHEL ROLLE (1652–1719) established that if a curve
intersects the x-axis at two locations a and b, is continuous, and has a TANGENT at every point between a and
Rolle’s theorem 449
eighth axiom holds:
8. Commutative Law for Multiplication: For all elements a and b of R, we have: a*b = b*a.
For example, the set of integers under ordinary addition
and multiplication is a commutative ring. Thus any
result that is known to follow abstractly from the eight
principles outlined above translates to a result about
numbers. The set of all functions f from the set of real
numbers to itself also satisfies the definition of a commutative ring, and so these results also translate to
interesting facts in function theory. As a simple example, it is straightforward to prove that the zero element
in a ring is unique. (If 0 and 0′ are both zeros, then, by
axiom three, 0 = 0 + 0′ = 0′.) Consequently, there is also
only one function that can behave as the zero function.
One can impose further conditions on a system.
For example, a commutative ring is called an integral
domain if a ninth axiom holds:
9. No Divisors of Zero: It is never the case that two
nonzero elements a and b of R give a*b = 0.
The set of integers is an integral domain, but the
set of functions is not. For example, if f is the function
that gives the value zero for negative inputs, and the
value 1 otherwise, and g is the function that gives the
value zero for positive inputs, and the value 1 otherwise, then neither f nor g is the zero function, but their
product f*g is. Any system in MODULAR ARITHMETIC
forms a ring but not necessarily an integral domain.
For example, 2 × 3 = 0 modulo 6.
A commutative ring is called a field if, further, a
10th axiom is satisfied:
10. Existence of Multiplicative Inverses: For each
nonzero element a of R, there is an element b such
that a*b = b*a = 1.
The set of integers is not a field. (The number 2,
for example, has no multiplicative inverse, since 1/2 is
not an integer.) The set of all real numbers under
addition and multiplication, however, is a field, as is
the set of all the rational numbers and the set of all
complex numbers.
It is usually assumed that the elements 0 and 1 in a
commutative ring or a field are different. If these two
elements are the same, that is, if 0 = 1, then one can
prove that the ring R contains only this element: R =
{0}. This ring is called the trivial ring.
See also ABSTRACT ALGEBRA; ASSOCIATIVE; COMMUTATIVE PROPERTY; DISTRIBUTIVE PROPERTY; GROUP;
GROUP THEORY; QUATERNIONS; ZERO.
Rolle, Michel (1652–1719) French Calculus Born on
April 21, 1652, in Ambert, France, Michel Rolle is best
remembered for the theorem in CALCULUS that bears his
name. Although attracting little attention at the time of
its publication, Rolle’s theorem is today considered one
of the fundamental principles of the subject.
Rolle began his career as a scribe and as an assistant to an attorney. He had little formal education but
pursued a personal interest in mathematics all his life.
He moved to Paris in 1675 and soon developed a reputation as an expert in arithmetic work. In 1682 he
achieved national fame for solving a recreational mathematics problem publicly posed by French mathematician Jacques Ozanam. In honor of this achievement,
Rolle was awarded a pension by France’s controller of
general finance and admission to the Académie Royal
des Sciences in 1685.
In 1690 he published the work Traité d’algèbre
(Treatise on algebra), a text on the theory of equations,
in which, among other things, he invented and used the
notation
n
√
–
x for the nth root of x. Rolle also studied
GEOMETRY, ALGEBRA, and DIOPHANTINE EQUATIONs.
His famous theorem was published one year later in an
obscure and little noticed text Démonstration d’une
méthode pour resoudre les égalitez de tous les degrez
(Proof of a method for solving equations of all degrees).
It is ironic that Rolle is today considered a principal figure in the development of calculus. Having studied the emerging subject, Rolle is said to have found
the theory unconvincing. He even went so far as to say
that calculus is nothing more than a “collection of
ingenious fallacies.” He died in Paris, France, on
November 8, 1719.
See also ROLLE’S THEOREM.
Rolle’s theorem In 1691 French mathematician
MICHEL ROLLE (1652–1719) established that if a curve
intersects the x-axis at two locations a and b, is continuous, and has a TANGENT at every point between a and
Rolle’s theorem 449