How to Give Every Rational Number a Room at Hilbert’s Hotel

For example. Let’s take the rational number  -3/4 .

In what room in Hilbert’s Hotell will we find that number?

Just insert the values -3 (=a) and 4 (=b) into the formula and we get…

so in room 1408 we’ll find -3/4.

Note that the comparison operators like ”<” or ”≥” evaluates to 1 if true or 0 if false.

This shows that the set of rational numbers is enumerable.

To represent every possible rational number it may seem unnecessary to allow for negative integers in both nominator and denominator… Yes, that’s true. It’ not necessary. I just wanted to keep the symmetry.



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