Dedekind CutsDate: 10/23/96 at 19:30:15 From: mat stern Subject: Dedekind cut I have already figured out Dedekind's theory of the rings and number notation. I still cannot figure out what his theory of the Dedekind cut is. Could you please send me a couple of sentences of what his cut is all about--so that a 7th grader can understand? I have contacted several mathamatic instructors and they know but cannot tell me in language that I can understand. Date: 10/24/96 at 10:22:13 From: Doctor Jerry Subject: Re: Dedekind cut Dedekind invented the idea of a Dedekind cut to show how the real numbers can be constructed from the rational numbers. The idea is that if you have the set of rational numbers (all numbers of the form p/q, where p and q are integers and q is not zero; the integers can be positive or negative), it ought to be possible to somehow create numbers like e and pi from this raw material. For example, Dedekind said that it would be possible to think of the number sqrt(2), which is not a rational number, as two sets of rational numbers. Let L (for left) be the set of all rational numbers whose squares are less than 2 and let R (for right) be the set of all rational numbers whose squares are bigger than 2. Both L and R can be entirely defined and described within the rational number system. We construct the number sqrt(2) by agreeing that sqrt(2) = {L,R}. This new object, a pair of sets, is what we will say is sqrt(2). The entire set of reals can be constructed by taking all possible pairs of subsets {L,R} of the rational numbers, where L and R must satisfy certain conditions (for example, every member of L must be less than every member of R). After creating the set of Dedekind cuts, one then defines how to add and multiply Dedekind cuts. There is a very nice book on this and related constructions. It's called Foundations of Analysis, by Edmund Landau. I have a copy published in 1951 by Chelsea Publishing Company. I see that I've written more than a couple of sentences. I hope that I've helped you understand a little about Dedekind cuts. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 11/25/96 at 15:23:57 From: Doctor Ceeks Subject: Re: Dedekind cut Hi, There came a time in the history of mathematics where people became very concerned that certain results were not properly proven. Mathematics can take pride in the absolute correctness of its results these days, but in the early days, many people made very loose arguments and "proved" some things which really make no sense. Even Leonard Euler provided examples of such nonsense. When people realized these loose arguments were becoming a real threat to the absolute correctness of mathematical theorems, some of them decided to try to develop mathematics very, very carefully. This meant rebuilding the counting numbers, and then the integers, and then the rational numbers very, very carefully to ensure that mathematics had a solid and secure foundation. After constructing the rational numbers, the next step was to construct the real numbers, of which pi, e, and the square root of 2, are non-rational examples. Have you ever wondered what exactly the real numbers are? You may think it is obvious, but there were many things which early mathematicians thought were obvious and based their arguments upon, only later, to find out that what they thought was obvious, was not even true. Examples of such things usually do not appear in high school mathematics because there, mathematical structures are simple enough that our intuition tends to guide us sure and true. In any case, the process of making mathematics "rigorous" included constructing in a systematic way, the real number system. Dedekind devised a way of constructing the real numbers from the rationals by utilizing the "Dedekind Cut". What Dedekind felt intuitively was that the set of rational numbers less than a given real number should determine that real number uniquely. One fact which Dedekind wanted to be true about real numbers was that between any two real numbers, there should be a rational number. In modern terminology, this means that Dedekind wanted the rational numbers to be "dense" amongst the real numbers. This fact is consistent with Dedekind's idea in the previous paragraph, because assuming this fact, one can see that if X and Y are different real numbers, then the set of rationals less than X must differ from the set of rationals less than Y, since there must be a rational number in between X and Y. Thus, Dedekind was led to make the following definition: Let R be the set of subsets S of the rational numbers with the following property: 1. S has no largest element. 2. If the rational number q is in S, then every rational number less than q must also be in S. Dedekind hoped that he could make R into a model of the real number system. To do this, he had to define a way of adding and multiplying elements of R. He found that he could do this, and that the resulting addition and multiplication satisfy all the laws we want the real numbers to satisfy. He then declared that R, along with the addition and multiplication he devised, be called the "Real Number System". Now, mathematicians can rest assured that the real number system exists. But the point is, if a mathematician is trying to prove something and needs to use a fact about real numbers which s/he isn't sure about, then all s/he has to do is see if s/he can deduce that the fact holds for the system R defined above. S/he doesn't have to make it a guessing game. Please write back if you have questions about the above. -Doctor Ceeks, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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