p +q

––––

2

140 discontinuity

toward solving the general problem since the time of

P

IERRE DE

F

ERMAT

(1601–65), who had established that

there are no solutions to the fourth-degree equation, and

L

EONHARD

E

ULER

(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 degree-

14 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, Dirich-

let 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

C

ARL

F

RIEDRICH

G

AUSS

(1777–1855) as chair of math-

ematics 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 EQUATION

s, 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 (Lec-

tures 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, Ger-

many. 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 exam-

ple, the set of

INTEGER

s 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 one-

to-one correspondence with the integers, a countable

set is usually regarded as discrete. This can be confus-

ing, 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

pand qlies 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

EVENT

s is called discrete if the underlying pop-

ulation 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 dis-

tance 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

ax2+ bx + c= 0 has solutions given by the quadratic

formula:

The quantity under the square root sign, b2–4ac, is

called the discriminant of the equation. If the discrimi-

nant of a quadratic is positive, then the equation has two

distinct real roots. For example, the equation 2x2–5x+ 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,

x2–6x+ 9 = 0, with discriminant zero, has only x= 3 as

a root. (It is a

DOUBLE ROOT

.) If the discriminant is nega-

tive, then the quadratic has no real solutions. It does,

however, have (distinct) complex solutions. For example,