Question
Mon April 25, 2011 By:

How can you prove that there are more irrational numbers than rational numbers between any set of two numbers?

Expert Reply
Mon April 25, 2011
While it is true that there are an infinite number of both rational and irrational numbers, you should know that, in a particular mathematical sense, there are way more irrational numbers than rational.

The reason for this is that there are two basic different types of 'infinite' sets - countably infinite and uncountably infinite.

Rationals comprise a countably infinite set. What this means is that you can assign an order among rationals, so you can say that this is the first and this is the second and so on. But of course, you will never end the process.

But the Irrationals are an uncountably infinite set. What this means is that you CANNOT even assign an order among irrationals. In this sense, there are more irrationals that rationals.

Another thing is that people say that both are equal since both are infinite. However infinite is not a number, so saying infinite = infinite is absurd.

You can read up more with at the page given in the reference.

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