R.DEDEKIND PRESENTED BY : ANIKET VISHWAKARMA

Richard Dedekind LIFE

Born - October 6, 1831 (1831-10-06) Braunschweig, Duchy of Brunswick

Died - February 12, 1916 (aged 84) (1916-02-13) Braunschweig, German EmpireNationality - German

Fields - Mathematician Philosopher of mathematicsDoctoral advisor - Carl Friedrich Gauss

Richard Dedekin

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Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory), algebraic number theory and the definition of the real numbers.

Richard

Dedekind life

Dedekind's father was Julius Levin Ulrich Dedekind, an administrator of Collegium Carolinum in Braunschweig. Dedekind had three older siblings.

As an adult, he never used the names Julius Wilhelm. He was born, lived most of his life, and died in Braunschweig (often called "Brunswick" in English).

Richard Dedekind life

He first attended the Collegium Carolinum in 1848 before transferring to the University of Göttingen in 1850. There, Dedekind was taught number theory by professor Moritz Stern. Gauss was still teaching, although mostly at an elementary level, and Dedekind became his last student. Dedekind received his doctorate in 1852, for a thesis titled Über die Theorie der Eulerschen Integrale ("On the Theory of Eulerian integrals"). This thesis did not display the talent evident by Dedekind's subsequent publications.

Richard Dedekind life

At that time, the University of Berlin, not Göttingen, was the main facility for mathematical research in Germany. Thus Dedekind went to Berlin for two years of study, where he and Bernhard Riemann were contemporaries; they were both awarded the habilitation in 1854. Dedekind returned to Göttingen to teach as a Privatdozent, giving courses on probability and geometry.

He studied for a while with Peter Gustav Lejeune Dirichlet, and they became good friends. Because of lingering weaknesses in his mathematical knowledge, he studied elliptic and abelian functions. Yet he was also the first at Göttingen to lecture concerning Galois theory.

About this time, he became one of the first people to understand the importance of the notion of groups for algebra and arithmetic

Richard Dedekind life

In 1858, he began teaching at the Polytechnic school in Zürich (now ETH Zürich). When the Collegium Carolinum was upgraded to a Technische Hochschule (Institute of Technology) in 1862, Dedekind returned to his native Braunschweig, where he spent the rest of his life, teaching at the Institute. He retired in 1894, but did occasional teaching and continued to publish. He never married, instead living with his sister Julia.

Dedekind was elected to the Academies of Berlin (1880) and Rome, and to the French Academy of Sciences (1900). He received honorary doctorates from the universities of Oslo, Zurich, and Braunschweig.

Richard Dedekindwork

While teaching calculus for the first time at the Polytechnic school, Dedekind developed the notion now known as a Dedekind cut (German: Schnitt), now a standard definition of the real numbers.

The idea of a cut is that an irrational number divides the rational numbers into two classes (sets), with all the numbers of one class (greater) being strictly greater than all the numbers of the other (lesser) class.

For example, the square root of 2 defines all the negative numbers and the numbers the squares of which are less than 2 into the lesser class, and the positive numbers the squares of which are greater than 2 into the greater class.

Every location on the number line continuum contains either a rational or an irrational number. Thus there are no empty locations, gaps, or discontinuities. Dedekind published his thoughts on irrational numbers and Dedekind cuts in his pamphlet "Stetigkeit und irrationale Zahlen" ("Continuity and irrational numbers");[1] in modern terminology, Vollständigkeit, completeness.

Rational and Real Numbers

The Rational Numbers are a field

Rational Numbers are an integral domain, since all fields are integral domains

Rational Order

Is (Q,+,•) an ordered integral domain? Recall the definition of ordered.

Ordered Integral Domain: Contains a subset D+ with the following properties.

1. If a, b D+ ,then a + b D+ (closure)2. If a , b D+ , then a • b D+ (closure)3. For each a Integral Domain D exactly

one of these holds a = 0, a D+, -a D+

(Trichotomy)

Rational Order

How can we define the positive set of rational numbers?

Verify closure of addition for the positive set Suppose show Verify closure of multiplication for the positive set Suppose show

Qdc

baQ

dcand

ba

Rational Order

Verify the Trichotomy LawIf a/b is a Rational Number then either a/b is positive, zero, or negative.

Dense Property

Between any two rational numbers r and s there is another rational number.

Determine a rule for finding a rational number between r and s. Verify it.

Rational Holes

Can any physical length be represented by a rational number?

Is the number line complete – does it still have gaps?

Pythagorean Society

Believed all physical distances could be represented as ratio of integers – our rational numbers.

500 B.C discovered the following : h2 = 12 + 12, h2 = 2, h = ? (not rational )

h11

Rational Incompleteness

Rational Numbers are sufficient for simple applications to physical problems

Theoretically the Rational Numbers are inadequate

Are these equations solvable over Q:

4x2 = 25 x2 = 13

Rational Incompleteness

Where does reside on the number line?

Are the Rational Numbers sufficient to complete the number line?

13x

3 43.5

Julius Wihelm Richard Dedekind

Born: 6 Oct 1831 in Braunschweig, (now Germany)Died: 12 Feb 1916 in Braunschweig

His idea was that every real number r divides the rational numbers into two subsets, namely those greater than r and those less than r.

Dedekind’s brilliant idea was to represent the real numbers by such divisions of the rationals.

Richard Dedekind

Richard Dedekind

Among other things, he provides a definition independent of the concept of number for the infiniteness or finiteness of a set by using the concept of mapping.

Presented a logical theory of number and of complete induction, presented his principal conception of the essence of arithmetic, and dealt with the role of the complete system of real numbers in geometry in the problem of the continuity of space.

Some quotes from Richard dedkind

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