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### Rational and Irrational Numbers

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Date: 11/12/97 at 20:56:15
From: Cinda Merrill
Subject: Rational and irrational numbers

Which set is bigger, the set of rational or irrational numbers?

I think they are the same because there are infinity numbers of both
but I cannot find a way to prove this statement.
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Date: 11/23/97 at 17:54:25
From: Doctor Allan
Subject: Re: Rational and irrational numbers

Hi Cinda,

The question you ask is very interesting but quite difficult to answer
in a few lines because the answer requires some rather sophisticated
set theory - but I will try my best, proceeding as follows: First I
will try to give an intuitive explanation and afterwards I will sketch
the theory needed in order to give a formal answer to your question.
I will put some references at the end of my answer.

You are in fact asking whether the rational and the irrational numbers
considered as sets have "the same number of elements." This is not a
new problem; in a letter dated 1873 the German mathematician Georg
Cantor asked another German mathematician Richard Dedekind the
following question:

Take the collection of all positive whole numbers n and denote it
by (n); then think of the collection of all real numbers x and
denote it by (x); the question is simply whether (n) and (x) may be
corresponded so that each individual of one collection corresponds
to one and only one of the other?...As much as I am inclined to the
opinion that (n) and (x) permit no such unique correspondence, I
cannot find the reason.

What Cantor is suggesting is that even though both the set of natural
numbers and the set of real numbers are infinite, they do not have
"the same number of elements." It was Cantor himself that proved the
impossibility of this correspondence and hence proved that the set of
real numbers is bigger than the set of natural numbers. He also proved
conjectures that will answer your question, and I will try to explain
his ideas.

Suppose we put all infinite sets into two different classes; one
consisting of the infinite sets whose elements we are able to count,
and the other one consisting of the infinite sets for which this isn't
possible. Since the natural numbers are also known as the counting
numbers it is reasonable to say that the natural numbers belong to the
first class of infinite sets, and almost the same reasoning would
yield that the integers should belong to this class also, because one
can think of the integers as twice the natural numbers - first you can
count the positive integers, and then you can count the negative
integers.

What about the rational numbers then? Is it possible in any way to
count the rational numbers? Yes it is; remember that any rational
number can be written as a fraction a/b, where a and b are integers
and b is not zero. So it should be possible to count the rational
numbers if you can count the integers.

What about the irrational numbers and the real numbers? Well, the real
numbers is the set consisting of all rational numbers and all
irrational numbers - so if we can't count the irrational numbers then
we can't count the real numbers (if we can't count the elements of a
subset of the real numbers then we definitely can't count the elements
of the set itself).

The main difference between the rational and irrational numbers in
this connection is that you can't find a correspondence between the
irrational numbers and the integers as it is the case with the
rational numbers. It is the very essence of an irrational number that
you can't write it as an integer fraction. So one would expect that
the irrational numbers and therefore the real numbers would belong to
the second class of infinite sets, and this is indeed the case. The
proof of this fact will be sketched later. But in regard to your
question this means that the set of irrational numbers is bigger than
the set of rational numbers.

I hope the above intuitive explanation gave you some idea of how it is
possible to say that one infinite set is larger than another one.
I will now go into detail with some of the mathematical terms trying
to give you an idea of how one can formalize the intuitive idea.

When you have two finite sets it is easy to tell which one is the
biggest. You simply count the elements of the sets, and the one with
most elements is the biggest. For infinite sets it is more subtle -
how do we count the number of elements when both sets are infinite?

Cantor's solution in his two papers published in 1895 and 1897 and
titled 'Beiträge zur Begründung der transfiniten Mengenlehre'
(Contributions to the Founding of Transfinite Set Theory) was to
introduce the concept of cardinality and the cardinal number. This was
meant to be an extension of the number of elements in a finite set and
should therefore work for finite sets also, which we will see that it
in fact does.

DEFINITION: Two sets A and B are called equivalent (A~B) if there is
a bijection f: A -> B.

A function is a bijection if it is surjective and injective. Let me
define these properties of a function.

DEFINITION: A function f is surjective if, whenever you take an
element of B, say b, I can take an element of A, say a, such that
f(a) = b.

This means that no matter what element of B you give me there will
exist at least one element of A that corresponds to the element of B.

EXAMPLE: Say A = {1,2,3,4} and B = {3,6,9,12} and f: A -> B is given
by f(x) = 3*x. Then f is surjective, because if you take some element
of B (9 for instance) then I can take an element a of A such that
f(a) = 9 - in this case a = 3, because f(3) = 3*3 = 9.

DEFINITION: A function f is injective if, whenever f(x) = f(y) then
x = y.

This means that no two elements of A will map to the same element,
meaning that there is exactly one element that maps to a given element
in the domain.

EXAMPLE: If A = {1,2,3,4}, B = {3,6,9,12} and f(x) = 3*x as in the
above example, then f is injective. You should try to check this
yourself.

So that f is a bijection between sets A and B means that an element of
one set corresponds to one and only one of the other set - and if you
look at the quotation from Cantor's letter, you will find that this is

DEFINITION: If there is a bijection between sets A and B then they
have the same cardinal number.

This definition will work for finite sets. A set M is finite if
there exists a natural number n such that M~(N_n+1), where
N_n+1 = {k in N: k < n+1}. So the cardinal number of a finite set will
just equal the number of elements in the set.

EXAMPLE: The cardinal number of the set A = {3,5,21} is 3 (having
A~N_4), while the cardinal number of the set B = {John, Cinda, Allan,
Bill} is 4. (B~N_5)

We know that the natural numbers N, the integers Z, the rational
numbers Q and the real numbers R are all infinite sets, and it is
possible to show that Z~N. In fact you will have to check that the
following function f: N -> Z is a bijection

f(n) = n/2       whenever n is even
f(n) = -(n-1)/2  whenever n is odd

In ordinary language I am saying that whenever you give me an element
of the integers I can give you exactly one element of the natural
numbers that corresponds to your element.

DEFINITION: We call a set M countable if it is finite or if M~N. If a
set is not countable it is called uncountable.

You see that an infinite set may be countable (meaning you can keep
track of the elements) - the only requirement is that there exists a
bijection from the set M to the natural numbers. Whenever a set is
countable it has the same cardinal number as the natural number, and
this cardinal number (which is the smallest infinite cardinal number)
is called aleph-zero.

THEOREM: The natural numbers and the rational numbers are equivalent,
meaning they have the same cardinal number (aleph-zero).

This can be shown using something called Cantor's diagonal method.
What Cantor actually did was finding a way of lining up the rational
numbers in a particular order and thereby being able to create the
necessary bijection. Using the above terminology we say that the
rational numbers are countable.

In order to answer your question we will have to examine whether the
irrational numbers are countable or uncountable. If they are
countable, then the rational and the irrational numbers have the same
cardinal number. If they aren't then the set containing the irrational
numbers is the bigger one.

We have to notice three things order to show that the irrational
numbers are uncountable.

THEOREM:

(1) Every countable union of countable sets is countable
(2) The real numbers R are uncountable
(3) R = (R/Q) union Q

To elaborate on (3): R/Q is just a short way of writing the set
consisting of those real numbers that are not rational numbers, which
is exactly the set of irrational numbers, and once you have the set of
irrational numbers and put the elements of this set together with the
rational numbers you will just get the real numbers again. (2) can be
proved using another diagonal argument of Cantor, and once you have
these three observations you can prove that the irrational numbers R/Q
are uncountable as follows:

THEOREM: The irrational numbers are uncountable

Proof: Assume that the set R/Q is countable.
We know that the set Q is countable, and (1) yields that R/Q union Q
is countable. By (3) we have that the set R is countable, which
contradicts (2). We must give up our assumption and may conclude that
R/Q is uncountable, yielding that the irrational numbers is the bigger
set.

I didn't get into details with the proofs here because I just wanted
to give you a general idea of what happens.

I hope that this made you interested in a more detailed treatment of
this topic. If that's the case you should consult for instance these
two books:

Halmos: Naive Set Theory
Rudin: Principles of Mathematical Analysis

Good luck!

-Doctor Allan,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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