toward solving the general problem since the time of
PIERRE DE FERMAT (1601–65), who had established that
there are no solutions to the fourth-degree equation, and
LEONHARD EULER (1707–83), who had proved that
there are no solutions to the third-degree equation.
Dirichlet was later able to extend his work to the degree14 equation, but to no other cases.
In honor of his achievement, Dirichlet was
awarded an honorary doctorate from the University of
Cologne, and with an advanced degree in hand, Dirichlet then pursued an academic career. He was appointed
professor of the University of Berlin in 1828, where he
remained for 27 years. In 1855 Dirichlet succeeded
CARL FRIEDRICH GAUSS (1777–1855) as chair of mathematics at the University of Göttingen.
Dirichlet developed innovative techniques using the
notion of a LIMIT in the study of NUMBER THEORY that
allowed him to make significant advances in the field.
He presented his famous result on ARITHMETIC
SEQUENCES to the Academy of Sciences on July 27,
1837, and published the work in the two-part paper
“Recherches sur diverses applications de l’analyse
infinitésimale à la théorie des nombres” (Inquiry on
various applications of infinitesimal analysis to number
theory) during the 2 years that followed. Dirichlet also
found applications of this work to mechanics, to the
solution of DIFFERENTIAL EQUATIONs, and to the study
of Fourier series. He consistently published papers on
both number theory and mathematical physics
throughout his career. His most notable works include
the 1863 book Vorlesungen über Zahlentheorie (Lectures on number theory), the 1846 article “Über die
Stabilität des Gleichgewichts” (On the stability of the
solar system), and the 1857 article “Untersuchungen
über ein Problem der Hydrodynamik” (Investigation
on a problem in hydrodynamics).
Dirichlet died on May 5, 1859, in Göttingen, Germany. Given the significance of his mathematical work,
many mathematicians of today regard Dirichlet as the
founder of analytic number theory.
discontinuity See CONTINUOUS FUNCTION.
discrete A set of numerical values in which there are
no intermediate values is said to be discrete. For example, the set of INTEGERs is discrete, but the set of all
REAL NUMBERS is not: between any two real numbers,
no matter how close, there is another real number. Any
finite set of values is considered discrete.
Since a COUNTABLE set of values can be put in oneto-one correspondence with the integers, a countable
set is usually regarded as discrete. This can be confusing, however, since the countable set of RATIONAL
NUMBERS, for instance, is discrete in this second sense,
but not in the first: between any two rational numbers
p and q lies another rational (
, for instance). One
must rely on the context of the problem under study to
determine whether or not the set of rational numbers
should be regarded as discrete.
In STATISTICS and PROBABILITY theory, a set of DATA
or set of EVENTs is called discrete if the underlying population is finite or countably infinite. The results of
tossing a die, for instance, form a discrete set of events,
since the die must land on one of six faces. In contrast,
for example the range of heights of Australian women
aged 36 is not discrete but continuous.
In GEOMETRY, an ISOMETRY with the property that
each point is moved more than some fixed positive distance further away is called a discrete transformation.
For example, a translation is discrete, but a rotation or
reflection is not.
Discrete geometry is the study of a finite set of
points, lines, circles, or other simple figures.
discriminant A QUADRATIC equation of the form
ax
2 + bx + c = 0 has solutions given by the quadratic
formula:
The quantity under the square root sign, b
2 – 4ac, is
called the discriminant of the equation. If the discriminant of a quadratic is positive, then the equation has two
distinct real roots. For example, the equation 2x
2 –5x + 2
= 0 has discriminant 3 and the two real solutions x = 2
and x = 1/2. If the discriminant of a quadratic is zero,
then the equation has just one real root. For instance,
x
2 –6x + 9 = 0, with discriminant zero, has only x = 3 as
a root. (It is a DOUBLE ROOT.) If the discriminant is negative, then the quadratic has no real solutions. It does,
however, have (distinct) complex solutions. For example,
x
b
b
ac
a
=
− ±
−
2
4
2
p +q
––––
2
140 discontinuity
PIERRE DE FERMAT (1601–65), who had established that
there are no solutions to the fourth-degree equation, and
LEONHARD EULER (1707–83), who had proved that
there are no solutions to the third-degree equation.
Dirichlet was later able to extend his work to the degree14 equation, but to no other cases.
In honor of his achievement, Dirichlet was
awarded an honorary doctorate from the University of
Cologne, and with an advanced degree in hand, Dirichlet then pursued an academic career. He was appointed
professor of the University of Berlin in 1828, where he
remained for 27 years. In 1855 Dirichlet succeeded
CARL FRIEDRICH GAUSS (1777–1855) as chair of mathematics at the University of Göttingen.
Dirichlet developed innovative techniques using the
notion of a LIMIT in the study of NUMBER THEORY that
allowed him to make significant advances in the field.
He presented his famous result on ARITHMETIC
SEQUENCES to the Academy of Sciences on July 27,
1837, and published the work in the two-part paper
“Recherches sur diverses applications de l’analyse
infinitésimale à la théorie des nombres” (Inquiry on
various applications of infinitesimal analysis to number
theory) during the 2 years that followed. Dirichlet also
found applications of this work to mechanics, to the
solution of DIFFERENTIAL EQUATIONs, and to the study
of Fourier series. He consistently published papers on
both number theory and mathematical physics
throughout his career. His most notable works include
the 1863 book Vorlesungen über Zahlentheorie (Lectures on number theory), the 1846 article “Über die
Stabilität des Gleichgewichts” (On the stability of the
solar system), and the 1857 article “Untersuchungen
über ein Problem der Hydrodynamik” (Investigation
on a problem in hydrodynamics).
Dirichlet died on May 5, 1859, in Göttingen, Germany. Given the significance of his mathematical work,
many mathematicians of today regard Dirichlet as the
founder of analytic number theory.
discontinuity See CONTINUOUS FUNCTION.
discrete A set of numerical values in which there are
no intermediate values is said to be discrete. For example, the set of INTEGERs is discrete, but the set of all
REAL NUMBERS is not: between any two real numbers,
no matter how close, there is another real number. Any
finite set of values is considered discrete.
Since a COUNTABLE set of values can be put in oneto-one correspondence with the integers, a countable
set is usually regarded as discrete. This can be confusing, however, since the countable set of RATIONAL
NUMBERS, for instance, is discrete in this second sense,
but not in the first: between any two rational numbers
p and q lies another rational (
, for instance). One
must rely on the context of the problem under study to
determine whether or not the set of rational numbers
should be regarded as discrete.
In STATISTICS and PROBABILITY theory, a set of DATA
or set of EVENTs is called discrete if the underlying population is finite or countably infinite. The results of
tossing a die, for instance, form a discrete set of events,
since the die must land on one of six faces. In contrast,
for example the range of heights of Australian women
aged 36 is not discrete but continuous.
In GEOMETRY, an ISOMETRY with the property that
each point is moved more than some fixed positive distance further away is called a discrete transformation.
For example, a translation is discrete, but a rotation or
reflection is not.
Discrete geometry is the study of a finite set of
points, lines, circles, or other simple figures.
discriminant A QUADRATIC equation of the form
ax
2 + bx + c = 0 has solutions given by the quadratic
formula:
The quantity under the square root sign, b
2 – 4ac, is
called the discriminant of the equation. If the discriminant of a quadratic is positive, then the equation has two
distinct real roots. For example, the equation 2x
2 –5x + 2
= 0 has discriminant 3 and the two real solutions x = 2
and x = 1/2. If the discriminant of a quadratic is zero,
then the equation has just one real root. For instance,
x
2 –6x + 9 = 0, with discriminant zero, has only x = 3 as
a root. (It is a DOUBLE ROOT.) If the discriminant is negative, then the quadratic has no real solutions. It does,
however, have (distinct) complex solutions. For example,
x
b
b
ac
a
=
− ±
−
2
4
2
p +q
––––
2
140 discontinuity