Zero is several sorts of numbers. It is actually both. "Real numbers", "Rational numbers", "Irrational Numbers", "integers", "trancendental numbers", "cardinal numbers" and so forth, all have their own definitions. These definitions define a SET of numbers. A single number can belong to several SETs.
Real numbers are numbers that can be written as an infinite decimal expansion. It includes numbers like fractions eg 1/7 and PI. Real numbers can also be split into algebraic and transcendental numbers but the definition is too complicated here. 0 is a member of this SET
Rational numbers work out to be the same but there is a slightly different definition. 0 is a member of this SET
Irrational numbers belong to the set of two dimensional numbers of the vector form a + b x i where i is the square root of -1. 0 = 0 + 0 x i so zero is also a member of this SET
Integers are the set of normal counting numbers both positive, negative and 0.
I have not included any more here because they are a part of number theory and not part of a discussion on logic (SETs).
No, it is a rational number.
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Such a product is always irrational - unless the rational number happens to be zero.
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.
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)
No. If the rational number is not zero, then such a product is irrational.
Provided that the rational number is not 0, the product is irrational.