{"id":573,"date":"2019-06-29T22:27:33","date_gmt":"2019-06-29T22:27:33","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=573"},"modified":"2019-11-18T23:20:08","modified_gmt":"2019-11-18T23:20:08","slug":"dedekind-cuts-of-rational-numbers","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/dedekind-cuts-of-rational-numbers\/","title":{"rendered":"Dedekind Cuts of Rational Numbers"},"content":{"rendered":"\n

Given a number line with equally spaced tick marks one unit apart, we know how to measure rational lengths: the length m\/n can be obtained by dividing a length m line segment into n equal parts (if you like, this can be done by straightedge and compass). A very natural question you might ask is whetherall<\/em> lengths on the line are rational length?<\/p>\n\n\n\n

The Greeks knew that this was not the case; the square root of two is in fact irrational and can be obtained as the hypotenuse of a right triangle with side lengths 1 and 1. And there are other lengths (like Pi) which are irrational, but cannot be constructed by straightedge and compass?<\/p>\n\n\n\n

These numbers (representing lengths) have an ordering, thus can be associated with points along a line. What the above remarks show is that the set rational numbers in this line has “gaps”. How does one “fill in the gaps” between the rational numbers?<\/p>\n\n\n\n

One way to do this was proposed by Dedekind in 1872, who suggested looking at “cuts”. A cut<\/em> C is a proper subset of rational numbers that is non-empty, has no greatest element, and is closed to the left (if r is in C, then any rational q < r is also in C).<\/p>\n\n\n\n

Cuts can be shown to have a natural ordering (by inclusion), a natural arithmetic, and in a very natural way “contain” an isomorphic copy of the rational numbers (the cut associated to a rational r is the set of all rationals less than r). But the set of cuts also contain uncountably many more elements. The set of all such cuts is called the real numbers<\/em>. In effect, we have constructed<\/em> the real numbers from the rationals!<\/p>\n\n\n\n

The Math Behind the Fact:<\/strong>
The technical details are best left to a course in real analysis. To add two cuts A and B, consider the set formed by summing one element of A with one element of B. Products may be defined similarly (but require one to be a little more careful). One can then show that the real numbers form a ordered field, and also satisfy the least upper bound property<\/em>: every non-empty subset that is bounded above has a least upper bound.<\/p>\n\n\n\n

This construction is one way to define<\/em> the real numbers. This set contains a cut that “behaves like” Sqrt[2], in that when you multiply it by itself, you get the cut corresponding to 2.<\/p>\n\n\n\n

How to Cite this Page:<\/strong>\u00a0
Su, Francis E., et al. “Dedekind Cuts of Rational Numbers.”\u00a0Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n

References:<\/strong>
Walter Rudin, Principles of Mathematical Analysis.<\/a><\/p>\n\n\n\n

Fun Fact suggested by:<\/strong>
Francis Su<\/p>\n","protected":false},"excerpt":{"rendered":"

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