synonymDedekind (Dedekin) Generally refers to Julius William Richard Dedekin
Julius Wilhelm Richard Dedekind (1831.10.6-1916.2.12) also translated Dedekin, a great German mathematician, theorist and educator, and a pioneer of modern abstract mathematics.According to《unabridged dictionary》DeDekin is also a Ph.DBerlin Academy academician.
DeDekin was born in Germany on October 6, 1831Lower SaxonyEastern CitiesBrunswickAn intellectual family.His father was a law professor, and his mother was also born in an intellectual family.In his early years, he studied chemistry and physics in a preparatory course at Brunswick University.[1]
DeDeJin when he was young.
In 1848, he entered Carolina College to study mechanicsCalculus, algebraic analysisanalytic geometryAnd natural science.
Transferred in in 1850University of GottingenThe new mathematics and physics seminar, from mathematician CF. Gauss studyleast square methodAnd advanced surveying, from Stern to the foundation of number theory, fromweberEngaged in physics, and took astronomy as an elective.
In 1852, with the title of "OnEulerThe theory of integral ".After graduation, he stayed as a substitute lecturer in 1854.[2]
The Pioneer of Abstract Mathematics -- Julius Dedekin
After the death of Gauss in 1855, DeDekin heard it again at the University of GottingenDirichletProfessor's number theory, potential theory, definite integral and partial differential equations, andBernhard Riemann He taught courses such as Abel function and elliptic function, and then came up with the idea of redefining irrational numbers with the aid of arithmetic properties.
Since 1855, he began to teach Galois theory and became the first scholar in the field of teaching.[2]
From 1858 to 1862, he was a professor at Zurich Institute of Technology.During this period, we mainly study the theoretical basis of real numbers.
From 1862 to 1912, he served as a professor at Brunswick Institute of Higher Technology, where he developed a continuous system of real numbers (without gaps) that can be composed of rational numbers and irrational numbers, provided that there is a one-to-one correspondence between real numbers and points on a straight line.He was successively elected as an academician of the French Academy of Sciences, the Berlin Academy of Sciences and the Rome Academy of Sciences.
Julius Dedekin
In 1888, DeDekin proposed a complete system of arithmetic axioms, includingMathematical inductionThe accurate expression of principle introduces many concepts of image into mathematics in the most common form.In addition, he also studied the basis of structure theory, making it one of the central branches of modern algebra.Many propositions and terms in modern mathematics, such as ring, field, structure, cross section, function, theorem, exchange principle, etc., are associated with his name.
On February 12, 1916, Dadkin died in Brunswick.Although many of his important ideas about the basic theory of mathematics were not fully understood before his death, they still affected the development of modern mathematics.
Achievements and honors
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Dai Dejin's main achievement is in algebra theory.He has studied any field, ring, group, structure, module and other issues, and was the first to introduce the concept of ring (field) in the course of teaching, and gave a general definition of ideal sub ring, and put forward the idea that the set that can establish a one-to-one correspondence with his own proper sub set is an infinite set.In the process of studying the theory of ideal subrings, he replaced the concept of ordered set (permutation group) with the concept of abstract group, and expressed it with a more common formula (the concept of DeDekin partition), which is much simpler than Cantor's formula, and directly affected the later birth of Piano's axiom of natural number.He is one of the first mathematicians who put forward many arguments for real number theory.Introduced in 1855 when teaching Galois theory“field'.[1-2]
DeDekin made many new discoveries in mathematics.Many concepts and theorems are named after him.His main contributions include the following two aspects: in the field of real number and continuity theory, he proposed“dedekind cut ”, givesIrrational numberAnd the definition of pure arithmetic of continuity.In 1872, his "Continuity and Irrational Numbers" was published, making him and Gcantor 、K.WeierstrassAnd become the founder of modern real number theory.In algebraic number theory, he established the modernAlgebraic numberandAlgebraic number fieldThe theory of EE.Kummer OfIdeal numberIt is popularized, and modern“ideal”Concept, and gotAlgebraic integer ringThe unique decomposition theorem of upper ideal.Today, the domain satisfying the ideal unique decomposition condition is called“Dai Dejin”。His contributions to number theory had a profound impact on mathematics in the 19th century.[1]
Main works
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Continuity and Irrational Numbers, Theory of Entire Algebra, Lectures on Number Theory, What is Number? What should Number be? And Mathematical Essays.
dedekind cut
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Given a certain method, divide all rational numbers into two sets, A and B. Each element in A is smaller than each element in B. Any classification method is called a partition of rational numbers.
For any segmentation, there must be three possibilities, and only one of them is true:
A has a maximum element a, and B has no minimum element.For example, A is all rational numbers ≤ 1, and B is all rational numbers ＞ 1.B has a minimum element b, and A has no maximum element.For example, A is all rational numbers<1.B is all rational numbers ≥ 1.A has no maximum element, and B has no minimum element.For example, A is all negative rational numbers, zero and square are less than 2 positive rational numbers, and B is all positive rational numbers with square greater than 2.Obviously, the union of A and B is all rational numbers, because the number whose square equals 2 is not rational.Note: It is impossible for A to have the largest element a and B to have the smallest element b, because then there is a rational number that does not exist in the two sets of A and B, and the union of A and B is the contradiction of all rational numbers.
In the third case, DeDekin said that the partition defined an irrational number, or simply said that the partition was an irrational number.[1]
In the first two cases, segmentation is rational.
In this way, all possible partitions constitute every point on the number axis, which has both rational and irrational numbers, collectively called real numbers.[3]
