Properties of Rational Number
Summary
Rational numbers :
Numbers that can be expressed in the form $\frac{\text{p}}{\text{q}}$, where p and q are integers and q ≠ 0, are known as rational numbers. The collection of rational numbers is denoted by Q.
These rational numbers satisfies various laws or properties that are listed below:
Rational numbers are closed under addition, subtraction and multiplication. If a, b are any two rational numbers, then and the sum, difference and product of these rational numbers is also a rational number, then we say that rational numbers satisfy the closure law.
Type of Number 
Closed Under  
Addition (+) 
Substraction () 
Multiplication (×) 
Division (÷) 

Whole Numbers  √  X  √  X 
Integers  √  √  √  X 
Rational Numbers  √  √  √  X 
Rational numbers are commutative under addition and multiplication. If a, b are rational numbers, then:
Commutative law under addition: a + b = b + a.
Commutative law under multiplication: a x b = b x a.
Type of Number 
Commutative Under  
Addition (+) 
Subtraction () 
Multiplication (×) 
Division (÷) 

Whole Numbers  √  X  √  X 
Integers  √  X  √  X 
Rational Numbers  √  X  √  X 
Rational numbers are associative under addition and multiplication. If a, b, c are rational numbers, then:
Associative law under addition: a + (b + c) = (a + b) + c
Associative law under multiplication: a(bc) = (ab)c
Type of Numbers 
Rational Numbers  
Addition (+) 
Substraction () 
Multiplication (x) 
Division (÷) 

Closure  √  X  √  X 
Commutative  √  X  √  X 
Associative  √  X  √  X 
• 0 is the additive identity for rational numbers.
• 1 is the multiplicative identity for rational numbers.
• The additive inverse of a rational number $\frac{\text{p}}{\text{q}}$ is  $\frac{\text{p}}{\text{q}}$, and the additive inverse of  $\frac{\text{p}}{\text{q}}$ is $\frac{\text{p}}{\text{q}}$.
• If $\frac{\text{p}}{\text{q}}$ x $\frac{\text{a}}{\text{b}}$ = 1, then $\frac{\text{a}}{\text{b}}$ is the reciprocal or multiplicative inverse of $\frac{\text{p}}{\text{q}}$ , and vice versa.
• For all rational numbers, p, q and r, p(q + r ) = pq + pr and p(q  r ) = pq  pr , is known as the distributive property.
Distribution property of multiplication over substraction
p(q  r) = pq  pr where p,q and r are rational numbers.
Distributive property of multiplication over addition
p(q + r) = pq + pr where p,q and r are rational numbers.
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Sol:
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