Thread: Finding rational and irrational between two numbers

My professor assigned a proof that between any two real numbers there exists both a rational and an irrational number. She told us to think about writing the rationals in the form of and write the irrationals in the form of , where n is rational.

Any ideas on how to write n in terms of two numbers, say a and b, such that n times its respective constant falls between a and b?

Originally Posted by Jameson

My professor assigned a proof that between any two real numbers there exists both a rational and an irrational number. She told us to think about writing the rationals in the form of and write the irrationals in the form of , where n is rational.

Any ideas on how to write n in terms of two numbers, say a and b, such that n times its respective constant falls between a and b?

Scrolling down this link might help get the ball rolling ...?

Here is a proof that I like very much. I first saw in it in Martin Davis’ Applied Nonstandard Analysis.
LEMMA: .
The proof of the is straight forward using properties of the floor function.
. Note that is an integer.

Suppose that then Because .
But this means or there is a rational between a & b.

If , then .
This means that .