If r is a non-zero rational number and s is irrational, what is rs?
Answer and explanation
Correct answer: Irrational number
The governing property is that the product of a non-zero rational number and an irrational number is irrational. Suppose, for contradiction, that rs were rational. Because r is non-zero and rational, its reciprocal 1/r is also rational. Multiplying the assumed rational number rs by 1/r would give s = (rs)(1/r), which would be rational. This contradicts the given fact that s is irrational. Therefore rs must be irrational. The condition r ≠ 0 is essential: if r were zero, the product would be 0, which is rational. It is not necessarily natural either, because the product may be negative or non-integral. Hence option A is correct.
Frequently asked questions
What is the correct answer to this question?
Irrational number
Why is this the correct answer?
The governing property is that the product of a non-zero rational number and an irrational number is irrational. Suppose, for contradiction, that rs were rational. Because r is non-zero and rational, its reciprocal 1/r is also rational. Multiplying the assumed rational number rs by 1/r would give s = (rs)(1/r), which would be rational. This contradicts the given fact that s is irrational. Therefore rs must be irrational. The condition r ≠ 0 is essential: if r were zero, the product would be 0, which is rational. It is not necessarily natural either, because the product may be negative or non-integral. Hence option A is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Irrational numbers and real numbers.
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