Rational NumbersDate: 11/24/97 at 15:04:02 From: Douglas Oliver Subject: Rational numbers Shady Grove and I frequently play with math questions together. She is in 6th grade (just skipped from 5th) and working on 8th grade math. A question came up when working on a problem from The Kaplan Edge. I can't explain the answer to her because it doesn't make sense to me. Here is the question and answer provided by Kaplan: Rational Observation Which is greater, the number of rational numbers between 0 and 1 or the number of rational numbers between 0 and 2? (A) the number of numbers between 0 and 1 (B) the number of numbers between 0 and 2 (C) there are no rational numbers in given ranges (D) the sets are the same size (E) can't be determined from information given The correct answer is: (D) Explanation: The rational numbers between 0 and 1 can be paired off systematically with the rational numbers between 0 and 2. Each number in the first range will be paired with twice that number in the second range. Because there is a one-to-one correspondence between the members of the two sets, you can say that the sets are the same size. Questions: Why are the numbers being paired in the first place? If the numbers between 0 and 1 don't include 1.75 for example, how can one say that the set of numbers between 0 and 1 is equal to the set of numbers between 0 and 2. The second set obviously, to me anyway, includes many more rational numbers, e.g. 1.75. What are we missing in this question? Or is the question simply a bad one? Thank you, Sincerely, Douglas and Shady Grove Oliver Date: 12/21/97 at 15:29:03 From: Doctor Naomi Subject: Re: rational numbers Dear Douglas and Shady Grove Oliver, Your question is a wonderful one, and we hope that the following answer is useful. Odd as it may seem, the answer given by Kaplan is correct! The answer probably goes against your intuition because the solution is based upon the arithmetic of transfinite, or infinite, sets. The first thing to notice is that there are an unlimited number of rational numbers between 0 and 1. For example, all of the numbers of the form 1/n, where n is a counting number 1, 2, 3...etc., are in this interval. Let us refer to the set of rational numbers between 0 and 1 as set A. Of course, there are also an infinite number of rational numbers between 0 and 2; we will call this set B. When comparing the number of elements in infinite sets such as these, mathematicians normally proceed by trying to find a rule (also called a "correspondence" or a "function") which relates each element in one set to an element in the other set. If such a rule can be set up, one which allows you to identify a corresponding number in both directions, then the sets are considered to be of the same (infinite) size. Remember that the original question asks you to compare the *size* of the two sets A and B, which means comparing the number of elements in sets A and B. In one of your questions you point out that A and B do not share all of the same elements and thus are not the same set. You are correct: set A does not equal set B (i.e., all of the elements in set B are not in set A); however, two sets that are not equal (that do not share all of the same elements) can still be of the same size, or cardinality (they can still have the same number of elements). In your example, the rule for pairing is that every number n in set A corresponds to the number 2n in B, and every number m in B corresponds to the number m/2 in A. Thus 0.3 in A corresponds to 0.6 in B, and 1.1 in B corresponds to 0.55 in A. Notice that the correspondence is symmetric - numbers in B have a counterpart in A, and vice versa. Let us formalize this a bit and call our rule a function f which maps points from set A into set B. So if n is a point in A, f(n) = 2n. Note that we can also move from elements in set B back to elements in set A using the function "f inverse": if m is a number in set B, "f inverse"(m) = m/2. This function is called f inverse because for every number n in set A, f(f inverse(n)) = n. What does this all mean in terms of our problem? Because it has an inverse, the function f is a bijection, which means (1) every element of B equals f(n) for some number n in A and (2) different numbers in A get mapped by f into different numbers in B. These properties might give you some intuition about why the existence of a bijective function f mapping between our sets causes them to be of the same size. In essence, the bijection tells us that for every distinct element in one of our sets, there is a corresponding distince element in the other set (no elements are missed or have more than one partner). This way of comparing infinite sets was devised because it turns out that there are infinite sets of very different sizes - so different that it is impossible to set up a bijective correspondence such as we did above. The sets A and B that we defined above are called "countably infinite," which basically means that we can put the rational numbers in order (1/1, 1/2, 1/3, ... , 2/1, 2/2, 2/3, ..., etc). The set of irrational numbers between 0 and 1 is also infinite, but it is *not* countably infinite like the rationals. Indeed this set of irrationals is so large that it is impossible to devise a way of putting the irrationals into a bijective correspondence with the rational numbers or any other countable set. These different "sizes" of infinity have been given names. The number of rational numbers in an interval, e.g. your set A, is called Aleph null. (Aleph is the first letter of the Hebrew alphabet). As you might guess, there are even higher orders of infinity beyond Aleph null, which you can read about at http://mathforum.org/dr.math/problems/mcswain11.7.97.html We know this is a confusing issue, but it is also fascinating. We hope our comments have helped you out a bit! -Doctors Roger and Naomi, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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