Numbers

Teaching the Real Number line is fascinating. How dense is it? Are there any holes?
I start off teaching the Natural or Counting numbers. These are easy. They are the numbers you naturally count with, 1, 2, 3, ... where the ... means keeps going to infinity. We can also have the opposites of these, the negative counting numbers,-1, -2, -3, ... off to negative infinity. We can also add in the number 0. It turns out 0,1,2,3... has a special namec alled the Whole Numbers. And if we take the whole numbers and negative counting numbers we get the Integers. So, now we have a nice set of numbers called the Integers that looks like
...-3,2,1,0,1,2,3...
But what's between 0 and 1? 1/2 or 0.5; 2/3 or 0.666... are examples. And, as before, their opposites or the negatives of these. So we can really pack them in. The Integers and all the stuff in between the integers.
The problem is numbers like 1/2, 3/4, 2/3 are very well behaved. They act rationally. As a matter of fact, they are called Rational Numbers. By definition, the Rational Numbers are numbers of the form p/q where p and q are Integers and q cannot be zero.What throws alot of people is that 5, for example, is a Rational Number because 5=5/1 and so fits the definition. An interesting property of the Rational Numbers is that their decimal versions either Terminate (3/4=.75) or Repeat (2/3=.666...)
So our Number line is filling up with the Integers and these things called Rational Numbers between the Integers (and actually including the Integers - as seen before 5=5/1)
What else is there? What about numbers whose decimal expansion do not repeat and do not terminate. For example, 3.1415926536897... These have a name also. They are the Irrational Numbers.
So finally, once we put in the Integers, Rational and Irrational numbers we end up with a verydense set of numbers, the Real Numbers that make up the Real Number line:
As an example ...-3,-2,-1,0,0.5,0.666..., 0.75,1,2,3,3.14159...,3.5,4... Hopefully you get the picture. Between any two numbers, Integer, Rational or Irrational, are more numbers.

There is an Infinite number of Integers
There is an Infinite number of numbers between any two numbers
There is an infinite number of even numbers {2,4,6,...}
There is an Infinite number of odd numbers {1,3,5,7...}
There are even special numbers called prime numbers divisible only by 1 and themseleves:{2,3,5,7,11,13,17...}
There is an infinite number of primes also.

Are we missing any numbers? or are we as dense as can be? Any holes?

You think about it.

Oh, one last thing. Are there any numbers whose digits are all primes? There must be beut how can we construct them?