From the Fun Fact files, here is a Random Fun Fact, at the Advanced level: |
Dedekind Cuts of Rational Numbers
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 whether
all lengths on the line are rational length?
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?
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?
One way to do this was proposed by Dedekind in 1872, who suggested looking at "cuts". A cut 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).
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.
In effect, we have constructed the real numbers from the rationals!
The Math Behind the Fact:
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:
every non-empty subset that is bounded above has a least upper bound.
This construction is one way to define the real numbers. This set contains a cut that "behaves like" Sqrt, in that when you multiply it by itself, you get the cut corresponding to 2.
How to Cite this Page:
Su, Francis E., et al. "Dedekind Cuts of Rational Numbers."
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