Subsets of Real Numbers and InfinityDate: 08/22/2001 at 17:26:39 From: Kevin Kelley Subject: Subsets of Real Numbers and Infinity Today in my Algebra II class we had a rather vehement argument, to which I have not been able to find a definitive answer. Both the Whole Number set and the Integer set have an infinite number of entries, yet some of my classmates persisted that there were more numbers in the Integer set because it includes negatives as well. Am I correct in saying that they both have an infinite number of numbers within them, and therefore are of the same size, or are my classmates correct in saying that the Integer set has more numbers in it? Date: 08/23/2001 at 10:34:22 From: Doctor Peterson Subject: Re: Subsets of Real Numbers and Infinity Hi, Kevin. You are right. Although there are in fact different infinities (for example, there are more real numbers than integers), the whole numbers and the integers have the same number of elements. (More surprisingly, so does the set of rational numbers.) You can see this by matching the two sets up: 0 1 2 3 4 5 6 7 8 ... 0 1 -1 2 -2 3 -3 4 -4 ... Since I can count the integers by matching each one up to a whole number, the two sets are the same size. You can read more about these ideas here: Large Numbers and Infinity - Dr. Math FAQ http://mathforum.org/dr.math/faq/faq.large.numbers.html Counting Rationals and Integers - Dr. Math archives http://mathforum.org/dr.math/problems/kelley.10.6.99.html Sets Containing an Infinite Number of Members - Dr. Math archives http://mathforum.org/dr.math/problems/kate2.3.98.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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