Question

Give an example of two irrational numbers whose sum and product both are rational.

Answer 1

Hint: Here, we need to know that rational numbers are those numbers that can be written into the form of \[\dfrac{p}{q}\] where \[p\] and \[q\] are integers. But irrational numbers are those which can not be written in the form of \[\dfrac{p}{q}\] . Below are the examples for both:
Rational number: \[\dfrac{1}{2}\], \[\dfrac{4}{5}\]…. etc.
Irrational number: \[\sqrt 3 \] , \[\sqrt 2 \] …. etc.

Complete step-by-step answer:
Step 1: Let's take two irrational numbers \[\sqrt 3 \] and \[ - \sqrt 3 \].
Step 2: Now, we will find the sum of these two irrational numbers as shown below:
\[{\text{Sum}} = \sqrt 3 + \left( { - \sqrt 3 } \right)\]
By opening the brackets into the RHS side of the above expression \[{\text{Sum}} = \sqrt 3 + \left( { - \sqrt 3 } \right)\], we get:
\[ \Rightarrow {\text{Sum}} = \sqrt 3 - \sqrt 3 \] ………….. (1)
By subtracting into the RHS side of the above expression (1), we get:
\[ \Rightarrow {\text{Sum}} = 0\] , which is a rational number.
Step 3: Now, we will calculate the product of these two irrational numbers \[\sqrt 3 \] and \[ - \sqrt 3 \] as shown below:
\[{\text{Product}} = \sqrt 3 \times \left( { - \sqrt 3 } \right)\]
By opening the brackets into the RHS side of the above expression \[{\text{Product}} = \sqrt 3 \times \left( { - \sqrt 3 } \right)\], we get:
\[ \Rightarrow {\text{Product}} = \sqrt 3 \times - \sqrt 3 \] ……….. (2)
By doing multiplication into the RHS side of the above expression (2), we get:
\[ \Rightarrow {\text{Product}} = - 3\] , which is again a rational number.
So, we can see that the sum and product of two irrational numbers can also be equal to a rational number.

The above example of two irrational numbers \[\sqrt 3 \] and \[ - \sqrt 3 \] proved that sum and product of two irrational numbers can also be equal to a rational number.

Note: Students needs to remember some important points about rational and irrational numbers as below:
The Sum of any two rational numbers is equal to a rational number.
The product of any two rational numbers equals a rational number.
The sum of a rational number with an irrational number is an irrational number.
The product of a rational number with an irrational number is an irrational number.
The sum of two irrational numbers is not always irrational.
The product of two irrational numbers is not always irrational.