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Let q be a non-zero rational and x be an irrational number.Suppose q*x = p where p is rational.

Then x = p/q.

Then, since the set of rational numbers is closed under division (by non-zero numbers), p/q is rational.

But that means that x is rational, which contradicts x being irrational.

Therefore the supposition that q*x is rational must be false

ie the product of a non-zero rational and an irrational cannot be rational.

Q: Why is the product of a non - zero rational number and an irrational number is irrational?

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Such a product is always irrational - unless the rational number happens to be zero.

Actually the product of a nonzero rational number and another rational number will always be rational.The product of a nonzero rational number and an IRrational number will always be irrational. (You have to include the "nonzero" caveat because zero times an irrational number is zero, which is rational)

Provided that the rational number is not 0, the product is irrational.

No. If the rational number is not zero, then such a product is irrational.

The product of two rational numbers, as in this example, is always RATIONAL.However, if you mean 10 x pi, pi is irrational; the product of a rational and an irrational number is ALWAYS IRRATIONAL, except for the special case in which the rational number is zero.

Related questions

Such a product is always irrational - unless the rational number happens to be zero.

Actually the product of a nonzero rational number and another rational number will always be rational.The product of a nonzero rational number and an IRrational number will always be irrational. (You have to include the "nonzero" caveat because zero times an irrational number is zero, which is rational)

Provided that the rational number is not 0, the product is irrational.

Not if the rational number is zero. In all other cases, the product is irrational.

The product of 0 and an irrational is 0 (a rational), the product of a non-zero rational and any irrational is always irrational.

No, but the only exception is if the rational number is zero.

It is usually irrational but it can be rational if the ration number in the pair is zero. So the correct answer is "either".

No. If the rational number is not zero, then such a product is irrational.

It is always FALSE.

When a rational numbers is divided by an irrational number, the answer is irrational for every non-zero rational number.

The product of two rational numbers, as in this example, is always RATIONAL.However, if you mean 10 x pi, pi is irrational; the product of a rational and an irrational number is ALWAYS IRRATIONAL, except for the special case in which the rational number is zero.

It is always true.