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### Dedekind Cuts

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Date: 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
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.

-Doctor Ceeks,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School History/Biography
High School Number Theory

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