Dedekind cut 119







numbers. Also, any rational function decomposes into

a sum of

PARTIAL FRACTIONS

. The process of G

AUSSIAN

ELIMINATION

shows that any square

MATRIX

Adecom-

poses into the product of a lower triangular matrix L

and an upper triangular matrix U. An example of such

an LU factorization is:

Any

VECTOR

decomposes into a sum of basis vectors.

For instance:

<3, 2, 1> = 3 <1, 0, 0> +2< 0, 1, 0> +< 0, 0, 1>

And in

GEOMETRY

, as any polygon can be divided into

triangles, one could say that all polygons “decompose”

into a union of triangles.

Studying the simpler pieces in the decomposition of

an object can lead to general results about the object.

For instance, knowing that the interior angles of a tri-

angle sum to 180°allows us to immediately deduce

that the interior angles of any quadrilateral (the union

of two triangles) sum to 360°, and that the interior

angles of any pentagon (the union of three triangles)

sum to 540°.

See also

FACTORIZATION

.

Dedekind, Julius Wilhem Richard (1831–1916)

German Analysis Born on October 6, 1831, in

Braunschweig, now a part of Germany, Richard

Dedekind is remembered for his elegant construction of

the

REAL NUMBER

system, which is based on an idea

today known as a D

EDEKIND CUT

. This work repre-

sented an important step in formalizing mathematics.

In particular, it offered the means to finally put

CALCU

-

LUS

on a sound mathematical footing.

Dedekind studied

NUMBER THEORY

and calculus at

the University of Göttingen. He earned a doctoral

degree in 1852 under the supervision of C

ARL

F

RIEDRICH

G

AUSS

(he was Gauss’s final pupil), and two

years later obtained a habilitation degree granting him

the right to be a member of the university faculty.

In 1858 Dedekind accepted a position at the Poly-

technikum in Zürich. Dedekind realized that the foun-

dations of calculus, in particular, the properties of the

real-number system on which calculus rests, were not

properly understood. When faced with the challenge of

teaching calculus to students at the Polytechnikum for

the first time, Dedekind decided not to sidestep the

issue, but rather develop an approach that would prop-

erly justify the principles of the subject to himself and

to his students. This is when the idea of a Dedekind cut

came to him.

Dedekind published the details of this construc-

tion several years later in his famous 1872 paper

“Continuity and Irrational Numbers.” This paper was

extremely well received and was admired not only for

the brilliant ideas it contained, but also for the man-

ner in which those ideas were detailed. Dedekind

exhibited a talent for explaining mathematical con-

cepts with exceptional clarity.

In 1862 Dedekind returned to his hometown to

accept a position at the Brunswick Polytechnikum. He

remained there for the rest of his life. He never married

and lived his life with one of his sisters, who also

remained unmarried.

Dedekind received many honors for his outstanding

work, including election to the Berlin Academy in 1880,

the Academy of Rome and the Académie des Sciences,

Paris, in 1900, as well as honorary doctorates from the

Universities of Zurich, Brunswick, and Oslo. Dedekind

died in Brunswick, Germany, on February 12, 1916.

Dedekind made a lasting impact on the modern

understanding of the real-number system. Most every

college-level course on the topic of the real numbers

will discuss in detail the issues Dedekind explored.

Dedekind cut During the 1800s it became clear to

mathematicians that in order to prove that

CALCULUS

is mathematically sound one needs to properly define

what is meant by a real number and, moreover, show

that the real number system is “complete,” in the

sense that no points are “missing” from it. This is par-

ticularly important for establishing the

EXTREME

-

VALUE THEOREM

, the

INTERMEDIATE

-

VALUE THEOREM

,

and the

MEAN

-

VALUE THEOREM

. All the key theorems

in calculus rely on these three results.

Although the

RATIONAL NUMBERS

Qare relatively

easy to define, the system of rationals is certainly not

complete: the square root of 2, for example, is not a

fraction and so is “missing” from the set of rationals.

The task of defining exactly what is meant by an irra-

tional number perplexed scholars for a very long time,