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Similarly, every cut of reals is identical to the cut produced by a specific real number (which can be identified as the smallest element of the B set). In other words, the number line where every real number is defined as a Dedekind cut of rationals is a complete continuum without any further gaps.
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. Real numbers can be constructed as Dedekind cuts of rational numbers. [7] [8]
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 , with all the numbers of one class (greater) being ...
Beginning with Richard Dedekind in 1858, several mathematicians worked on the definition of the real numbers, including Hermann Hankel, Charles Méray, and Eduard Heine, leading to the publication in 1872 of two independent definitions of real numbers, one by Dedekind, as Dedekind cuts, and the other one by Georg Cantor, as equivalence classes ...
Dedekind completeness is the property that every Dedekind cut of the real numbers is generated by a real number. In a synthetic approach to the real numbers, this is the version of completeness that is most often included as an axiom. The rational number line Q is not Dedekind complete. An example is the Dedekind cut
If is the set of rational numbers, viewed as a totally ordered set with the usual numerical order, then each element of the Dedekind–MacNeille completion of may be viewed as a Dedekind cut, and the Dedekind–MacNeille completion of is the total ordering on the real numbers, together with the two additional values .
The definition of real numbers as Dedekind cuts was first published by Richard Dedekind in 1872. [32] The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 1?" by Fred Richman in Mathematics Magazine. [12]
Real numbers can be constructed as Dedekind cuts of rational numbers. For convenience we may take the lower set as the representative of any given Dedekind cut (,), since completely determines . By doing this we may think intuitively of a real number as being represented by the set of all smaller rational numbers.