The real number system

The simplest of numbers are of course the natural numbers (or those we use to count with), they are
1, 2, ....., 100, 101, .....
There are an infinite amount of these numbers, where the next one is the current number plus 1.
We also have the whole number, which are all the natural numbers, together with 0,
0, 1, 2, ....., 100, 101, .....
Then we get the integers, which include all the natural numbers and we add the negatives. Hence
..., -101, -100, ..., -2, -1, 0, 1, 2, ..., 100, 101, ....
Following this we have to rational numbers, which are all the numbers which we can express in terms of fractions,
a/b
where both the numerator a and denominator b are integers, which means that the rational numbers also include all integers, since 2 = 1/2 for example.
Irrational numbers are all those which have decimal places, but cannot be expressed in terms of a fraction, which means they have an infinite amount of decimals which doesn't repeat, for example
34, 76812........
Numbers which do have repetition, e.g. 2,65656565..... can be expressed in terms of a fraction and are therefore classified as rational.
The rational and irrational numbers combined form the real number system. Between any two numbers we can find another number, which is also included in this system.

What is left out from this system is the non-real, or imaginary numbers, but they form a whole new system of their own.