Question

Give an example for each of the following:
1.The product of two irrational numbers is a rational number.
2.The product of two irrational numbers is an irrational number.

Answer 1

Hint: Rational numbers are categorized as all numbers except irrational numbers. Irrational numbers are imaginary numbers. But we can obtain different types of numbers by applying various operations in mathematics such as multiplication and division. Hence, we can proceed by this methodology.

Complete step by step answer:
In mathematics, the number system is the branch that deals with various types of numbers possible to form and easy to operate with different operators such as addition, multiplication and so on.
Four major types of numbers can be classified as:
Natural numbers, whole numbers, integers, and rational numbers. Irrational numbers come under the category of imaginary numbers.
So, rational numbers are those numbers which can be represented in the form of $\dfrac{p}{q}$, where p is the numerator and q is the denominator. P and Q may belong to any of the three categories such as natural, whole or integers.
And, an irrational number is any real number other than a rational number.
As given in our problem, we have to find a pair of irrational numbers whose product can generate rational numbers and irrational numbers separately.
So, for forming a rational number:
Consider these two-irrational numbers: $\sqrt{2}\text{ and }\sqrt{8}$.
The product of these numbers is: $\sqrt{16}=4$ which is a rational number.
 So, for forming an irrational number:
Consider these two-irrational number: $\sqrt{2}\text{ and }\sqrt{\dfrac{3}{2}}$
The product of these numbers is: $\sqrt{2\times \dfrac{3}{2}}=\sqrt{3}$ which is an irrational number.
Hence, we obtained our solution for both parts.

Note: The key step for solving this problem is the knowledge of the number system and particularly rational and irrational numbers. The basic idea of a rational and irrational number system is good enough to solve both the parts. This knowledge is helpful in solving complex problems.