Countability of Rational and Irrational NumbersDate: 01/30/2001 at 14:00:57 From: Wade Whitehead Subject: Countability of rational and irrational numbers I asked this question on a test: "When speaking of countability of numbers, which has more - rational or irrational? Explain your answer." I feel it is irrational, because you may take every rational number and divide it by pi, or put pi as the numerator. Is my thinking correct? Help would be greatly appreciated. Thanks. Date: 01/30/2001 at 16:14:48 From: Doctor Schwa Subject: Re: Countability of rational and irrational numbers It's true that there are more irrationals than rationals, but your argument doesn't prove it. One way to think about it is that the rationals are an integer on top, and an integer on the bottom, so if Z is the number of integers, then Z^2 is how many rationals there are. On the other hand, since an irrational has a decimal expansion, with one decimal place for each integer, and each decimal place can be any value 0-9, there are 10^Z irrationals (well, that includes rationals too, but they are negligible by comparison, as we'll see). Now, if you think about the growth of the function x^2 as compared with 10^x, when x is big, 10^x is MUCH greater. So when x is infinite, it's REALLY greater. That's still not a completely rigorous mathematical argument. Especially when it comes to infinities, it's easy to slip into fallacies. So I recommend reading the *real* proof that there are more irrationals than rationals, involving the Cantor diagonalization methods, in our Frequently Asked Questions page: Large Numbers and Infinity http://mathforum.org/dr.math/faq/faq.large.numbers.html Enjoy, - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/ |
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