Rational and Irrational NumbersDate: 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/ |
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