Natural Numbers
The numbers 1, 2, 3… are called natural numbers or counting numbers.
Whole Numbers
All the whole numbers including 0, i.e. 0, 1, 2 ……. are whole numbers.
Integers (z)
All the whole numbers including positive as well as negative are integers.
Z = {5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5}
Rational Numbers
A number is called rational number if it can be expressed in the form of \(\frac{p}{q}\) where p and q are integers (q > 0)
Example: \(\frac{1}{2}\), \(\frac{4}{3}\), 1 etc.
Properties of Rational Numbers
Closure Property
The closure property states that when multiplication, subtraction etc.) on any two numbers in a certain type of number, the result of computation is another number in the same type of number.

Addition 
Subtraction 
Multiplication 
Division 
Rational Numbers 
Yes 
Yes 
Yes 
No 
Integers 
Yes 
Yes 
Yes 
No 
Whole Numbers 
Yes 
No 
Yes 
No 
Natural Numbers 
Yes 
No 
Yes 
No 
Commutative Property
The commutative property status that order does not matter. So, if you swap numbers in an operation (such as addition, multiplication, subtraction etc.) on any two numbers in a certain type of number, the result of the computation is same as the result without swap.

Addition 
Subtraction 
Multiplication 
Division 
Rational Numbers 
Yes 
No 
Yes 
No 
Integers 
Yes 
No 
Yes 
No 
Whole Numbers 
Yes 
No 
Yes 
No 
Natural Numbers 
Yes 
No 
Yes 
No 
Associative Property
The associative property states that it doesn't matter how we group the numbers. We can add parenthesis anywhere and we get the same answer.

Addition 
Subtraction 
Multiplication 
Division 
Rational Numbers 
Yes 
No 
Yes 
No 
Integers 
Yes 
No 
Yes 
No 
Whole Numbers 
Yes 
No 
Yes 
No 
Natural Numbers 
Yes 
No 
Yes 
No 
Additive Identity/ Role of Zero
a + 0 = 0 + a = a, where a is a whole number.
b + 0 = 0 + b = b, where b is an integer.
c + 0 = 0 + c = c, where c is a rational number.
Zero is called the identity for the addition of rational numbers. It is the additive identity for integers and whole numbers as well.
Multiplicative identity/ Role of one
We observe,
a \(\times\) 1 = a, where a is a whole number.
b \(\times\) 1 = b, where b is an integer.
c \(\times\) 1 = c, where c is a rational number.
1 is the multiplicative identity for rational numbers. It is the multiplicative identity for integers and whole numbers as well.
Additive Inverse
a + (a) = 0, where a is a whole number.
b + (b) = 0, where b is an integer.
(\(\frac{a}{b}\)) + (\(\frac{a}{b}\)) = 0, where \(\frac{a}{b}\) is a rational number.
So, we say that (\(\frac{a}{b}\)) is the additive inverse of \(\frac{a}{b}\) and \(\frac{a}{b}\) is the additive inverse of (\(\frac{a}{b}\)).
Reciprocal or Multiplicative Inverse
For all rational numbers a, b and c
 a(b + c) = ab + bc
 a(b – c) = ab – bc