Which set of numbers does the density property apply to real rational natural irrational or whole?

Answer 1

The rationals are dense since between any two there is always another. You can always add them and divide by 2. For example 1/2 and 1/3. You can add them and divide by 2 . 3/6+ 2/6=5/6 and half of that is 5/12 ( 5/12 is certainly between 4/12 and 6/12) The whole numbers are not dense. Is there a whole number between 1 and 2? I don't think so! And irrationals are dense as well, you can do the same thing you did with the rationals. Just add them and divide by 2. Of course there are many other numbers between each rational and each irrational, the idea of adding and dividing by two just ensures the existence of at least one such number. Now if the density property applies to rationals and irrationals, it must apply to reals since they can be viewed as the intersection of these two sets.