I'm not entirely sure I understand your comment, but the fact that there are more real numbers than natural numbers can be readily shown using something called cantor's diagonal argument. It goes something like this:

Suppose the set of real numbers and natural numbers had the same size. Then we could write down an infinite list, where each line represents some real number written in it's decimal representation. So something like

1: 3.14159265
2: 1.41421356
3: 0.24242424
...

This list goes on forever. We will now construct a new real number r as follows: The first number after the decimal point of r shall be different from the first number after the decimal point of the first number in our list, the second shall be different from the second decimal of the second number on the list, and so on (the name diagonal argument comes from this, we consider the entries on the diagonal from top left towards the bottom right).

The key point now is that this constructed real number is different from every single number on the list: After all, we have made sure it differs from each number on the list in at least one place. Therefore, it is impossible to write down the real numbers in such a way that each real number gets its own natural number: There are simply too few natural numbers for this. In particular, there are at least two different "sizes" of infinities.