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.

PS:
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.

Annonser

Kommentera

Fyll i dina uppgifter nedan eller klicka på en ikon för att logga in:

WordPress.com Logo

Du kommenterar med ditt WordPress.com-konto. Logga ut / Ändra )

Twitter-bild

Du kommenterar med ditt Twitter-konto. Logga ut / Ändra )

Facebook-foto

Du kommenterar med ditt Facebook-konto. Logga ut / Ändra )

Google+ photo

Du kommenterar med ditt Google+-konto. Logga ut / Ändra )

Ansluter till %s