Prove that there exist equal number of irrational numbers between any 2 rational numbers, when the difference between the 2 rational numbers is same.

Question

Prove that there exist equal number of irrational numbers between any 2 rational numbers, when the difference between the 2 rational numbers is same.

If the assertion is not true then please prove otherwise.

Answer 1

There's a nice proof of a much stronger theorem which is that there exist equal numbers of irrational numbers between any 2 distinct real numbers (even if the difference between the two numbers isn't the same).

To prove this, consider 2 line segments in the cartesian plane s1 and s2. The first is from P1=(s_1, 0) to P2=(e_1, 0), and the other from P3=(s_2, 1) to P4=(e_2, 1). Draw lines L1 and L2 which connect P1 to P3 and P2 to P4 respectively, and label their intersection P5. Now for any point x on s1 we can draw the line connecting P1 to x. This line intersects s2 at exactly one point, and it's fairly simple to prove that this forms a bijection between s1 and s2.