Simon, Asia, and Jiovanni are finding the product of a rational and irrational number.

Question

Simon says the result should be irrational because the product of an irrational and rational number is always irrational.

Asia says the result should be zero because the product of an irrational and rational number is always zero.

Jiovanni says the result should be rational because the product of an irrational and rational number is always rational.

Who is correct?

(1 point)
Responses

Simon
Simon

Asia
Asia

None of them are correct.
None of them are correct.

Jiovanni

Answer 1

Simon's reasoning is correct. The product of a non-zero rational number and an irrational number is always irrational. However, if the rational number happens to be zero, then the product will be zero, which is rational.

Asia's statement only applies when the rational number is zero, not in general. If the rational number is not zero, the product will be irrational.

Jiovanni is incorrect since the product of an irrational number and a non-zero rational number is not rational.

Therefore, the correct response is:

None of them are correct.

Answer 2

wrong

Answer 3

I apologize for the confusion. Let’s clarify the statements.

1. Simon is correct in saying that if a rational number \( r \) (where \( r \neq 0 \)) is multiplied by an irrational number \( x \), the product \( r \times x \) is irrational.

2. Asia is incorrect because she claims that the product is always zero, which would only be true if the rational number is zero. If the rational number is non-zero, the product will not be zero.

3. Jiovanni is incorrect because he says the product is always rational. This is not true since the product of a non-zero rational number and an irrational number is irrational.

Given this analysis:

• Simon is correct if the rational number is non-zero.
• Asia is correct only if the rational number is zero.
• Jiovanni is incorrect.

If the context implies solely the comparison of general cases without exceptions, then you could say Simon is the one who makes the correct statement about the typical case.

Thus, the best conclusion is that Simon is the correct answer regarding the general non-zero case, which is a common understanding.

Again, I apologize for the oversight.