Each set of numbers includes the numbers of the nested sets. For example, all natural numbers are whole numbers and all whole numbers are integers.
Natural Numbers: 1 2 3 4 etc (the numbers you count with)
Whole Numbers: Includes all Natural numbers, but adds 0 (That's a zero!) 0 1 2 3 etc
Integers: Includes all Natural numbers and all Whole numbers, but also adds negative numbers …. -4 -3 -2 -1 0 1 2 3 etc
Rational Numbers: Includes all of the above, but adds the fractions and decimals. ALL fractions are included and decimals that can be written as ending or repeating decimals. For example: ¼ and ½ and 3.5 and 3.33
Irrational Numbers: This is the first group that doesn't include the other numbers already defined. This is the group that includes non-ending or non-repeating decimals. Examples of this includes where x is not a perfect square and π (pi)
Real Numbers: Includes all of the above and doesn't add anything new, but it is the group that puts irrational numbers into the main group with the other numbers.
Now, the last group of numbers is called Complex or Imaginary Numbers. The most famous of these is i otherwise known as the square root of -1. There are a few others that are used in engineering and physics. This group is outside of the other groups above and sits all alone.