Theory:

Real numbers are either rational or irrational
 
We learned that rational numbers satisfies the commutative, associative and distributive properties for addition and multiplication. Thus rational number is closed under addition and multiplication.
 
We also learned that irrational numbers also satisfy the commutative, associative and distributive properties for addition and multiplication. 
 
But the sum, difference, product and division of two irrational numbers is not always irrational. The operations on irrational numbers results in rational or irrational.
 
Thus, it can be concluded that the irrational number is not closed under addition, subtraction, multiplication and division.
Let us recall some properties on irrationals from the previous topic and solve some problems on it.
  1. Addition, subtraction, multiplication and division of two irrational number is may or may not be irrational.
  2. Addition of rational and irrational number is always irrational.
  3. Subtraction of rational and irrational number is always irrational.
  4. Multiplication of rational and irrational is always irrational.
  5. Division of rational and irrational is always irrational.