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If a is any definite number, then all numbers of the system R fall into two classes, A<sub>1</sub> and A<sub>2</sub>, each of which contains infinite… - Richard Dedekind
" "If a is any definite number, then all numbers of the system R fall into two classes, A<sub>1</sub> and A<sub>2</sub>, each of which contains infinitely many individuals; the first class A<sub>1</sub> comprises all numbers a<sub>1</sub> that are < a, the second class A<sub>2</sub> comprises all numbers a<sub>2</sub> that are > a; the number a itself may be assigned at pleasure to the first or second class, being respectively the greatest number of the first class or the least of the second. In every case the separation of the system R into the two classes A<sub>1</sub>, A<sub>2</sub> is such that every number of the first class A<sub>1</sub> is less than every number of the second class A<sub>2</sub>.
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About Richard Dedekind
(6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to (particularly ), and the definition of the s.
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Native Name:
Julius Wilhelm Richard Dedekind
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The way in which the irrational numbers are usually introduced is based directly upon the conception of extensive magnitudes—which itself is nowhere carefully defined—and explains number as the result of measuring such a magnitude by another of the same kind. Instead of this I demand that arithmetic shall be developed out of itself.
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Just as negative and fractional rational numbers are formed by a new creation, and as the laws of operating with these numbers must and can be reduced to the laws of operating with positive integers, so we must endeavor completely to define irrational numbers by means of the rational numbers alone. The question only remains how to do this.
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