Natural numbers: from 1 upwards
Whole numbers: from 0 upwards
Integers: ...-3, -2, -1, 0, 1, 2, 3, ...
Rational numbers: fractions between two integers
Irrational numbers: can't be written as fraction between two integers
Real numbers: either can or cannot be written as fraction between two integers
These are all definitions, so there really isn't much to explain..They are just names we've given to these sets!