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There are two basic points behind Dedekind’s definition of a real number: (1) the geometric intuition that any real number divides the set of all real numbers into two halves, those smaller and those bigger; (2) and real number can be approximated arbitrarily well by rational numbers.
Dedekind Cuts and Real Numbers. DEFINITION 1.2.1. A Dedekind cut is a subset a of the rational numbers with the following properties: 1. a is not empty and a = 6 ; Q. 2. if p 2 a and q < p, then q 2 a; 3. if p 2 a, then there is some r 2 a such that r > p (i.e., a has no maximal element). DEFINITION 1.2.2.
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In mathematics, the Dedekind numbers are a rapidly growing sequence of integers named after Richard Dedekind, who defined them in 1897. The Dedekind number M(n) is the number of monotone boolean functions of n variables.
The short, simple answer used in calculus courses is that a real number is a point on the number line. That's not the whole truth, but it is adequate for the needs of freshman calculus.
Construction by Dedekind cuts. Dedekind used his cut to construct the irrational, real numbers. A Dedekind cut in an ordered field is a partition of it, ( A, B ), such that A is nonempty and closed downwards, B is nonempty and closed upwards, and A contains no greatest element.
How would you define a real number? It would seem that the easiest way is to say that a real number is a decimal expansion of the form. N.d1d2d3..., where N is one of 0, 1, 2, 3, ... and each digit dk is one of 0, ..., 9. You could then say that the above decimal expansion represents the number. d1 d2 d3. + + + ...
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How does Dedekind create real numbers?
How did Dedekind construct irrational numbers?
Why was Dedekind a great mathematician?
What are Dedekind numbers?
Apr 22, 2008 · As noted, Dedekind starts with the system of rational numbers; he uses a set-theoretic procedure to construct, in a central step, the new system of cuts out of them; and finally, the real numbers are “created” on that basis.
