Rational and Irrational Numbers

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.

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 
precisely what he asked for.

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/