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Countability of Rational and Irrational Numbers


Date: 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/   
    
Associated Topics:
High School Number Theory

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