Product of a Non-Zero Rational Number and an Irrational Number

Video Explanation

Watch the video below for a clear explanation:

Solution

Question: The product of a non-zero rational number and an irrational number is:

(a) always irrational    (b) always rational    (c) rational or irrational    (d) one

Key Concept

A rational number can be written in the form p/q, where p and q are integers and q ≠ 0.

An irrational number cannot be expressed in the form p/q.

Logical Explanation

Let r be a non-zero rational number and x be an irrational number.

Assume that the product r × x is rational.

Then:

x = (r × x) ÷ r

Since a rational number divided by a non-zero rational number is rational, this would make x rational.

But this contradicts the fact that x is irrational.

Hence, the assumption is wrong.

Final Answer

✔ The product of a non-zero rational number and an irrational number is always irrational.

✔ Correct option: (a) always irrational

Conclusion

Thus, multiplying any non-zero rational number with an irrational number always gives an irrational number.